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Theorem cofucl 13811
Description: The composition of two functors is a functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofucl.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
cofucl.g  |-  ( ph  ->  G  e.  ( D 
Func  E ) )
Assertion
Ref Expression
cofucl  |-  ( ph  ->  ( G  o.func  F )  e.  ( C  Func  E
) )

Proof of Theorem cofucl
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2316 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
2 cofucl.f . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
3 cofucl.g . . . 4  |-  ( ph  ->  G  e.  ( D 
Func  E ) )
41, 2, 3cofuval 13805 . . 3  |-  ( ph  ->  ( G  o.func  F )  =  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) >. )
51, 2, 3cofu1st 13806 . . . 4  |-  ( ph  ->  ( 1st `  ( G  o.func 
F ) )  =  ( ( 1st `  G
)  o.  ( 1st `  F ) ) )
64fveq2d 5567 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G  o.func 
F ) )  =  ( 2nd `  <. ( ( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) >. ) )
7 fvex 5577 . . . . . . 7  |-  ( 1st `  G )  e.  _V
8 fvex 5577 . . . . . . 7  |-  ( 1st `  F )  e.  _V
97, 8coex 5253 . . . . . 6  |-  ( ( 1st `  G )  o.  ( 1st `  F
) )  e.  _V
10 fvex 5577 . . . . . . 7  |-  ( Base `  C )  e.  _V
1110, 10mpt2ex 6240 . . . . . 6  |-  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )  e.  _V
129, 11op2nd 6171 . . . . 5  |-  ( 2nd `  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) >. )  =  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )
136, 12syl6eq 2364 . . . 4  |-  ( ph  ->  ( 2nd `  ( G  o.func 
F ) )  =  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) )
145, 13opeq12d 3841 . . 3  |-  ( ph  -> 
<. ( 1st `  ( G  o.func 
F ) ) ,  ( 2nd `  ( G  o.func 
F ) ) >.  =  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) >. )
154, 14eqtr4d 2351 . 2  |-  ( ph  ->  ( G  o.func  F )  =  <. ( 1st `  ( G  o.func 
F ) ) ,  ( 2nd `  ( G  o.func 
F ) ) >.
)
16 eqid 2316 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
17 eqid 2316 . . . . . . 7  |-  ( Base `  E )  =  (
Base `  E )
18 relfunc 13785 . . . . . . . 8  |-  Rel  ( D  Func  E )
19 1st2ndbr 6211 . . . . . . . 8  |-  ( ( Rel  ( D  Func  E )  /\  G  e.  ( D  Func  E
) )  ->  ( 1st `  G ) ( D  Func  E )
( 2nd `  G
) )
2018, 3, 19sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 1st `  G
) ( D  Func  E ) ( 2nd `  G
) )
2116, 17, 20funcf1 13789 . . . . . 6  |-  ( ph  ->  ( 1st `  G
) : ( Base `  D ) --> ( Base `  E ) )
22 relfunc 13785 . . . . . . . 8  |-  Rel  ( C  Func  D )
23 1st2ndbr 6211 . . . . . . . 8  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2422, 2, 23sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
251, 16, 24funcf1 13789 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
26 fco 5436 . . . . . 6  |-  ( ( ( 1st `  G
) : ( Base `  D ) --> ( Base `  E )  /\  ( 1st `  F ) : ( Base `  C
) --> ( Base `  D
) )  ->  (
( 1st `  G
)  o.  ( 1st `  F ) ) : ( Base `  C
) --> ( Base `  E
) )
2721, 25, 26syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( 1st `  G
)  o.  ( 1st `  F ) ) : ( Base `  C
) --> ( Base `  E
) )
285feq1d 5416 . . . . 5  |-  ( ph  ->  ( ( 1st `  ( G  o.func 
F ) ) : ( Base `  C
) --> ( Base `  E
)  <->  ( ( 1st `  G )  o.  ( 1st `  F ) ) : ( Base `  C
) --> ( Base `  E
) ) )
2927, 28mpbird 223 . . . 4  |-  ( ph  ->  ( 1st `  ( G  o.func 
F ) ) : ( Base `  C
) --> ( Base `  E
) )
30 eqid 2316 . . . . . . 7  |-  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )
31 ovex 5925 . . . . . . . 8  |-  ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  e.  _V
32 ovex 5925 . . . . . . . 8  |-  ( x ( 2nd `  F
) y )  e. 
_V
3331, 32coex 5253 . . . . . . 7  |-  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) )  e.  _V
3430, 33fnmpt2i 6235 . . . . . 6  |-  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )  Fn  ( (
Base `  C )  X.  ( Base `  C
) )
3513fneq1d 5372 . . . . . 6  |-  ( ph  ->  ( ( 2nd `  ( G  o.func 
F ) )  Fn  ( ( Base `  C
)  X.  ( Base `  C ) )  <->  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) ) )
3634, 35mpbiri 224 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G  o.func 
F ) )  Fn  ( ( Base `  C
)  X.  ( Base `  C ) ) )
372adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  F  e.  ( C  Func  D
) )
383adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  G  e.  ( D  Func  E
) )
39 simprl 732 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
40 simprr 733 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
411, 37, 38, 39, 40cofu2nd 13808 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  ( G  o.func 
F ) ) y )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )
42 eqid 2316 . . . . . . . . . . . 12  |-  (  Hom  `  D )  =  (  Hom  `  D )
43 eqid 2316 . . . . . . . . . . . 12  |-  (  Hom  `  E )  =  (  Hom  `  E )
4420adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  G ) ( D  Func  E )
( 2nd `  G
) )
4525adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) : ( Base `  C
) --> ( Base `  D
) )
4645, 39ffvelrnd 5704 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  x )  e.  ( Base `  D
) )
4745, 40ffvelrnd 5704 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  F
) `  y )  e.  ( Base `  D
) )
4816, 42, 43, 44, 46, 47funcf2 13791 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) ) : ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) --> ( ( ( 1st `  G ) `
 ( ( 1st `  F ) `  x
) ) (  Hom  `  E ) ( ( 1st `  G ) `
 ( ( 1st `  F ) `  y
) ) ) )
49 eqid 2316 . . . . . . . . . . . 12  |-  (  Hom  `  C )  =  (  Hom  `  C )
5024adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
511, 49, 42, 50, 39, 40funcf2 13791 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  F
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) ) )
52 fco 5436 . . . . . . . . . . 11  |-  ( ( ( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) ) : ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) --> ( ( ( 1st `  G ) `
 ( ( 1st `  F ) `  x
) ) (  Hom  `  E ) ( ( 1st `  G ) `
 ( ( 1st `  F ) `  y
) ) )  /\  ( x ( 2nd `  F ) y ) : ( x (  Hom  `  C )
y ) --> ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )  ->  (
( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) : ( x (  Hom  `  C )
y ) --> ( ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) (  Hom  `  E )
( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) ) )
5348, 51, 52syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) : ( x (  Hom  `  C )
y ) --> ( ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) (  Hom  `  E )
( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) ) )
54 ovex 5925 . . . . . . . . . . 11  |-  ( ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) (  Hom  `  E )
( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) )  e.  _V
55 ovex 5925 . . . . . . . . . . 11  |-  ( x (  Hom  `  C
) y )  e. 
_V
5654, 55elmap 6839 . . . . . . . . . 10  |-  ( ( ( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) )  e.  ( ( ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) (  Hom  `  E )
( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) )  ^m  ( x (  Hom  `  C )
y ) )  <->  ( (
( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) : ( x (  Hom  `  C )
y ) --> ( ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) (  Hom  `  E )
( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) ) )
5753, 56sylibr 203 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) )  e.  ( ( ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) (  Hom  `  E )
( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) )  ^m  ( x (  Hom  `  C )
y ) ) )
581, 37, 38, 39cofu1 13807 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  ( G  o.func 
F ) ) `  x )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) )
591, 37, 38, 40cofu1 13807 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  ( G  o.func 
F ) ) `  y )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) )
6058, 59oveq12d 5918 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( 1st `  ( G  o.func 
F ) ) `  x ) (  Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) )  =  ( ( ( 1st `  G ) `  (
( 1st `  F
) `  x )
) (  Hom  `  E
) ( ( 1st `  G ) `  (
( 1st `  F
) `  y )
) ) )
6160oveq1d 5915 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( ( 1st `  ( G  o.func  F )
) `  x )
(  Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) )  ^m  ( x (  Hom  `  C ) y ) )  =  ( ( ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) (  Hom  `  E )
( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) )  ^m  ( x (  Hom  `  C )
y ) ) )
6257, 61eleqtrrd 2393 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) )  e.  ( ( ( ( 1st `  ( G  o.func 
F ) ) `  x ) (  Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) )  ^m  ( x (  Hom  `  C ) y ) ) )
6341, 62eqeltrd 2390 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  ( G  o.func 
F ) ) y )  e.  ( ( ( ( 1st `  ( G  o.func 
F ) ) `  x ) (  Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) )  ^m  ( x (  Hom  `  C ) y ) ) )
6463ralrimivva 2669 . . . . . 6  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) ( x ( 2nd `  ( G  o.func 
F ) ) y )  e.  ( ( ( ( 1st `  ( G  o.func 
F ) ) `  x ) (  Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) )  ^m  ( x (  Hom  `  C ) y ) ) )
65 fveq2 5563 . . . . . . . . 9  |-  ( z  =  <. x ,  y
>.  ->  ( ( 2nd `  ( G  o.func  F )
) `  z )  =  ( ( 2nd `  ( G  o.func  F )
) `  <. x ,  y >. ) )
66 df-ov 5903 . . . . . . . . 9  |-  ( x ( 2nd `  ( G  o.func 
F ) ) y )  =  ( ( 2nd `  ( G  o.func 
F ) ) `  <. x ,  y >.
)
6765, 66syl6eqr 2366 . . . . . . . 8  |-  ( z  =  <. x ,  y
>.  ->  ( ( 2nd `  ( G  o.func  F )
) `  z )  =  ( x ( 2nd `  ( G  o.func 
F ) ) y ) )
68 vex 2825 . . . . . . . . . . . 12  |-  x  e. 
_V
69 vex 2825 . . . . . . . . . . . 12  |-  y  e. 
_V
7068, 69op1std 6172 . . . . . . . . . . 11  |-  ( z  =  <. x ,  y
>.  ->  ( 1st `  z
)  =  x )
7170fveq2d 5567 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( ( 1st `  ( G  o.func  F )
) `  ( 1st `  z ) )  =  ( ( 1st `  ( G  o.func 
F ) ) `  x ) )
7268, 69op2ndd 6173 . . . . . . . . . . 11  |-  ( z  =  <. x ,  y
>.  ->  ( 2nd `  z
)  =  y )
7372fveq2d 5567 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( ( 1st `  ( G  o.func  F )
) `  ( 2nd `  z ) )  =  ( ( 1st `  ( G  o.func 
F ) ) `  y ) )
7471, 73oveq12d 5918 . . . . . . . . 9  |-  ( z  =  <. x ,  y
>.  ->  ( ( ( 1st `  ( G  o.func 
F ) ) `  ( 1st `  z ) ) (  Hom  `  E
) ( ( 1st `  ( G  o.func  F )
) `  ( 2nd `  z ) ) )  =  ( ( ( 1st `  ( G  o.func 
F ) ) `  x ) (  Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) ) )
75 fveq2 5563 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( (  Hom  `  C ) `  z
)  =  ( (  Hom  `  C ) `  <. x ,  y
>. ) )
76 df-ov 5903 . . . . . . . . . 10  |-  ( x (  Hom  `  C
) y )  =  ( (  Hom  `  C
) `  <. x ,  y >. )
7775, 76syl6eqr 2366 . . . . . . . . 9  |-  ( z  =  <. x ,  y
>.  ->  ( (  Hom  `  C ) `  z
)  =  ( x (  Hom  `  C
) y ) )
7874, 77oveq12d 5918 . . . . . . . 8  |-  ( z  =  <. x ,  y
>.  ->  ( ( ( ( 1st `  ( G  o.func 
F ) ) `  ( 1st `  z ) ) (  Hom  `  E
) ( ( 1st `  ( G  o.func  F )
) `  ( 2nd `  z ) ) )  ^m  ( (  Hom  `  C ) `  z
) )  =  ( ( ( ( 1st `  ( G  o.func  F )
) `  x )
(  Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) )  ^m  ( x (  Hom  `  C ) y ) ) )
7967, 78eleq12d 2384 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( ( ( 2nd `  ( G  o.func 
F ) ) `  z )  e.  ( ( ( ( 1st `  ( G  o.func  F )
) `  ( 1st `  z ) ) (  Hom  `  E )
( ( 1st `  ( G  o.func 
F ) ) `  ( 2nd `  z ) ) )  ^m  (
(  Hom  `  C ) `
 z ) )  <-> 
( x ( 2nd `  ( G  o.func  F )
) y )  e.  ( ( ( ( 1st `  ( G  o.func 
F ) ) `  x ) (  Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) )  ^m  ( x (  Hom  `  C ) y ) ) ) )
8079ralxp 4864 . . . . . 6  |-  ( A. z  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ( ( 2nd `  ( G  o.func 
F ) ) `  z )  e.  ( ( ( ( 1st `  ( G  o.func  F )
) `  ( 1st `  z ) ) (  Hom  `  E )
( ( 1st `  ( G  o.func 
F ) ) `  ( 2nd `  z ) ) )  ^m  (
(  Hom  `  C ) `
 z ) )  <->  A. x  e.  ( Base `  C ) A. y  e.  ( Base `  C ) ( x ( 2nd `  ( G  o.func 
F ) ) y )  e.  ( ( ( ( 1st `  ( G  o.func 
F ) ) `  x ) (  Hom  `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  y ) )  ^m  ( x (  Hom  `  C ) y ) ) )
8164, 80sylibr 203 . . . . 5  |-  ( ph  ->  A. z  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ( ( 2nd `  ( G  o.func 
F ) ) `  z )  e.  ( ( ( ( 1st `  ( G  o.func  F )
) `  ( 1st `  z ) ) (  Hom  `  E )
( ( 1st `  ( G  o.func 
F ) ) `  ( 2nd `  z ) ) )  ^m  (
(  Hom  `  C ) `
 z ) ) )
82 fvex 5577 . . . . . 6  |-  ( 2nd `  ( G  o.func  F )
)  e.  _V
8382elixp 6866 . . . . 5  |-  ( ( 2nd `  ( G  o.func 
F ) )  e.  X_ z  e.  (
( Base `  C )  X.  ( Base `  C
) ) ( ( ( ( 1st `  ( G  o.func 
F ) ) `  ( 1st `  z ) ) (  Hom  `  E
) ( ( 1st `  ( G  o.func  F )
) `  ( 2nd `  z ) ) )  ^m  ( (  Hom  `  C ) `  z
) )  <->  ( ( 2nd `  ( G  o.func  F ) )  Fn  ( (
Base `  C )  X.  ( Base `  C
) )  /\  A. z  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ( ( 2nd `  ( G  o.func 
F ) ) `  z )  e.  ( ( ( ( 1st `  ( G  o.func  F )
) `  ( 1st `  z ) ) (  Hom  `  E )
( ( 1st `  ( G  o.func 
F ) ) `  ( 2nd `  z ) ) )  ^m  (
(  Hom  `  C ) `
 z ) ) ) )
8436, 81, 83sylanbrc 645 . . . 4  |-  ( ph  ->  ( 2nd `  ( G  o.func 
F ) )  e.  X_ z  e.  (
( Base `  C )  X.  ( Base `  C
) ) ( ( ( ( 1st `  ( G  o.func 
F ) ) `  ( 1st `  z ) ) (  Hom  `  E
) ( ( 1st `  ( G  o.func  F )
) `  ( 2nd `  z ) ) )  ^m  ( (  Hom  `  C ) `  z
) ) )
85 eqid 2316 . . . . . . . . . 10  |-  ( Id
`  C )  =  ( Id `  C
)
86 eqid 2316 . . . . . . . . . 10  |-  ( Id
`  D )  =  ( Id `  D
)
8724adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( 1st `  F ) ( C 
Func  D ) ( 2nd `  F ) )
88 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
891, 85, 86, 87, 88funcid 13793 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  F
) x ) `  ( ( Id `  C ) `  x
) )  =  ( ( Id `  D
) `  ( ( 1st `  F ) `  x ) ) )
9089fveq2d 5567 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  x
) ) `  (
( x ( 2nd `  F ) x ) `
 ( ( Id
`  C ) `  x ) ) )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  x
) ) `  (
( Id `  D
) `  ( ( 1st `  F ) `  x ) ) ) )
91 eqid 2316 . . . . . . . . 9  |-  ( Id
`  E )  =  ( Id `  E
)
9220adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( 1st `  G ) ( D 
Func  E ) ( 2nd `  G ) )
9325ffvelrnda 5703 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
9416, 86, 91, 92, 93funcid 13793 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  x
) ) `  (
( Id `  D
) `  ( ( 1st `  F ) `  x ) ) )  =  ( ( Id
`  E ) `  ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) ) )
9590, 94eqtrd 2348 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  x
) ) `  (
( x ( 2nd `  F ) x ) `
 ( ( Id
`  C ) `  x ) ) )  =  ( ( Id
`  E ) `  ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) ) )
962adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  F  e.  ( C  Func  D ) )
973adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  G  e.  ( D  Func  E ) )
981, 96, 97, 88, 88cofu2nd 13808 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( x
( 2nd `  ( G  o.func 
F ) ) x )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  x
) )  o.  (
x ( 2nd `  F
) x ) ) )
9998fveq1d 5565 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  ( G  o.func 
F ) ) x ) `  ( ( Id `  C ) `
 x ) )  =  ( ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  x
) )  o.  (
x ( 2nd `  F
) x ) ) `
 ( ( Id
`  C ) `  x ) ) )
1001, 49, 42, 87, 88, 88funcf2 13791 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( x
( 2nd `  F
) x ) : ( x (  Hom  `  C ) x ) --> ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  x
) ) )
101 funcrcl 13786 . . . . . . . . . . . . 13  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
1022, 101syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
103102simpld 445 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  Cat )
104103adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  C  e.  Cat )
1051, 49, 85, 104, 88catidcl 13633 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  C ) `  x )  e.  ( x (  Hom  `  C
) x ) )
106 fvco3 5634 . . . . . . . . 9  |-  ( ( ( x ( 2nd `  F ) x ) : ( x (  Hom  `  C )
x ) --> ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
)  /\  ( ( Id `  C ) `  x )  e.  ( x (  Hom  `  C
) x ) )  ->  ( ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  x
) )  o.  (
x ( 2nd `  F
) x ) ) `
 ( ( Id
`  C ) `  x ) )  =  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  x )
) `  ( (
x ( 2nd `  F
) x ) `  ( ( Id `  C ) `  x
) ) ) )
107100, 105, 106syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  x
) )  o.  (
x ( 2nd `  F
) x ) ) `
 ( ( Id
`  C ) `  x ) )  =  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  x )
) `  ( (
x ( 2nd `  F
) x ) `  ( ( Id `  C ) `  x
) ) ) )
10899, 107eqtrd 2348 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  ( G  o.func 
F ) ) x ) `  ( ( Id `  C ) `
 x ) )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  x
) ) `  (
( x ( 2nd `  F ) x ) `
 ( ( Id
`  C ) `  x ) ) ) )
1091, 96, 97, 88cofu1 13807 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( G  o.func  F ) ) `  x )  =  ( ( 1st `  G ) `  (
( 1st `  F
) `  x )
) )
110109fveq2d 5567 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  E ) `  ( ( 1st `  ( G  o.func 
F ) ) `  x ) )  =  ( ( Id `  E ) `  (
( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) ) )
11195, 108, 1103eqtr4d 2358 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x ( 2nd `  ( G  o.func 
F ) ) x ) `  ( ( Id `  C ) `
 x ) )  =  ( ( Id
`  E ) `  ( ( 1st `  ( G  o.func 
F ) ) `  x ) ) )
11287adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
113 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  x  e.  ( Base `  C
) )
114 simprlr 739 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  z  e.  ( Base `  C
) )
1151, 49, 42, 112, 113, 114funcf2 13791 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
x ( 2nd `  F
) z ) : ( x (  Hom  `  C ) z ) --> ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  z
) ) )
116 eqid 2316 . . . . . . . . . . . . 13  |-  (comp `  C )  =  (comp `  C )
117104adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  C  e.  Cat )
118 simprll 738 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  y  e.  ( Base `  C
) )
119 simprrl 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  f  e.  ( x (  Hom  `  C ) y ) )
120 simprrr 741 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  g  e.  ( y (  Hom  `  C ) z ) )
1211, 49, 116, 117, 113, 118, 114, 119, 120catcocl 13636 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
g ( <. x ,  y >. (comp `  C ) z ) f )  e.  ( x (  Hom  `  C
) z ) )
122 fvco3 5634 . . . . . . . . . . . 12  |-  ( ( ( x ( 2nd `  F ) z ) : ( x (  Hom  `  C )
z ) --> ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  z )
)  /\  ( g
( <. x ,  y
>. (comp `  C )
z ) f )  e.  ( x (  Hom  `  C )
z ) )  -> 
( ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  z
) )  o.  (
x ( 2nd `  F
) z ) ) `
 ( g (
<. x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  z
) ) `  (
( x ( 2nd `  F ) z ) `
 ( g (
<. x ,  y >.
(comp `  C )
z ) f ) ) ) )
123115, 121, 122syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  z )
)  o.  ( x ( 2nd `  F
) z ) ) `
 ( g (
<. x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  z
) ) `  (
( x ( 2nd `  F ) z ) `
 ( g (
<. x ,  y >.
(comp `  C )
z ) f ) ) ) )
124 eqid 2316 . . . . . . . . . . . . 13  |-  (comp `  D )  =  (comp `  D )
1251, 49, 116, 124, 112, 113, 118, 114, 119, 120funcco 13794 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( x ( 2nd `  F ) z ) `
 ( g (
<. x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( y ( 2nd `  F ) z ) `
 g ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) )
126125fveq2d 5567 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  z
) ) `  (
( x ( 2nd `  F ) z ) `
 ( g (
<. x ,  y >.
(comp `  C )
z ) f ) ) )  =  ( ( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  z
) ) `  (
( ( y ( 2nd `  F ) z ) `  g
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) ) )
127 eqid 2316 . . . . . . . . . . . 12  |-  (comp `  E )  =  (comp `  E )
12892adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  ( 1st `  G ) ( D  Func  E )
( 2nd `  G
) )
12993adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( 1st `  F
) `  x )  e.  ( Base `  D
) )
13025adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( 1st `  F ) : (
Base `  C ) --> ( Base `  D )
)
131130adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  ( 1st `  F ) : ( Base `  C
) --> ( Base `  D
) )
132131, 118ffvelrnd 5704 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( 1st `  F
) `  y )  e.  ( Base `  D
) )
133131, 114ffvelrnd 5704 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( 1st `  F
) `  z )  e.  ( Base `  D
) )
1341, 49, 42, 112, 113, 118funcf2 13791 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
x ( 2nd `  F
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) ) )
135134, 119ffvelrnd 5704 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( x ( 2nd `  F ) y ) `
 f )  e.  ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) ) )
1361, 49, 42, 112, 118, 114funcf2 13791 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
y ( 2nd `  F
) z ) : ( y (  Hom  `  C ) z ) --> ( ( ( 1st `  F ) `  y
) (  Hom  `  D
) ( ( 1st `  F ) `  z
) ) )
137136, 120ffvelrnd 5704 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( y ( 2nd `  F ) z ) `
 g )  e.  ( ( ( 1st `  F ) `  y
) (  Hom  `  D
) ( ( 1st `  F ) `  z
) ) )
13816, 42, 124, 127, 128, 129, 132, 133, 135, 137funcco 13794 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  z
) ) `  (
( ( y ( 2nd `  F ) z ) `  g
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) )  =  ( ( ( ( ( 1st `  F
) `  y )
( 2nd `  G
) ( ( 1st `  F ) `  z
) ) `  (
( y ( 2nd `  F ) z ) `
 g ) ) ( <. ( ( 1st `  G ) `  (
( 1st `  F
) `  x )
) ,  ( ( 1st `  G ) `
 ( ( 1st `  F ) `  y
) ) >. (comp `  E ) ( ( 1st `  G ) `
 ( ( 1st `  F ) `  z
) ) ) ( ( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) ) `  (
( x ( 2nd `  F ) y ) `
 f ) ) ) )
139123, 126, 1383eqtrd 2352 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  z )
)  o.  ( x ( 2nd `  F
) z ) ) `
 ( g (
<. x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( ( ( 1st `  F ) `  y
) ( 2nd `  G
) ( ( 1st `  F ) `  z
) ) `  (
( y ( 2nd `  F ) z ) `
 g ) ) ( <. ( ( 1st `  G ) `  (
( 1st `  F
) `  x )
) ,  ( ( 1st `  G ) `
 ( ( 1st `  F ) `  y
) ) >. (comp `  E ) ( ( 1st `  G ) `
 ( ( 1st `  F ) `  z
) ) ) ( ( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) ) `  (
( x ( 2nd `  F ) y ) `
 f ) ) ) )
14096adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  F  e.  ( C  Func  D
) )
14197adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  G  e.  ( D  Func  E
) )
1421, 140, 141, 113, 114cofu2nd 13808 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
x ( 2nd `  ( G  o.func 
F ) ) z )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  z
) )  o.  (
x ( 2nd `  F
) z ) ) )
143142fveq1d 5565 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( x ( 2nd `  ( G  o.func  F )
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  z
) )  o.  (
x ( 2nd `  F
) z ) ) `
 ( g (
<. x ,  y >.
(comp `  C )
z ) f ) ) )
144109adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( 1st `  ( G  o.func 
F ) ) `  x )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) )
1451, 140, 141, 118cofu1 13807 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( 1st `  ( G  o.func 
F ) ) `  y )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) )
146144, 145opeq12d 3841 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  <. (
( 1st `  ( G  o.func 
F ) ) `  x ) ,  ( ( 1st `  ( G  o.func 
F ) ) `  y ) >.  =  <. ( ( 1st `  G
) `  ( ( 1st `  F ) `  x ) ) ,  ( ( 1st `  G
) `  ( ( 1st `  F ) `  y ) ) >.
)
1471, 140, 141, 114cofu1 13807 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( 1st `  ( G  o.func 
F ) ) `  z )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  z ) ) )
148146, 147oveq12d 5918 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  ( <. ( ( 1st `  ( G  o.func 
F ) ) `  x ) ,  ( ( 1st `  ( G  o.func 
F ) ) `  y ) >. (comp `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  z ) )  =  ( <. ( ( 1st `  G ) `  (
( 1st `  F
) `  x )
) ,  ( ( 1st `  G ) `
 ( ( 1st `  F ) `  y
) ) >. (comp `  E ) ( ( 1st `  G ) `
 ( ( 1st `  F ) `  z
) ) ) )
1491, 140, 141, 118, 114, 49, 120cofu2 13809 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( y ( 2nd `  ( G  o.func  F )
) z ) `  g )  =  ( ( ( ( 1st `  F ) `  y
) ( 2nd `  G
) ( ( 1st `  F ) `  z
) ) `  (
( y ( 2nd `  F ) z ) `
 g ) ) )
1501, 140, 141, 113, 118cofu2nd 13808 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
x ( 2nd `  ( G  o.func 
F ) ) y )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )
151150fveq1d 5565 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( x ( 2nd `  ( G  o.func  F )
) y ) `  f )  =  ( ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) `
 f ) )
152 fvco3 5634 . . . . . . . . . . . . 13  |-  ( ( ( x ( 2nd `  F ) y ) : ( x (  Hom  `  C )
y ) --> ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
)  /\  f  e.  ( x (  Hom  `  C ) y ) )  ->  ( (
( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) `
 f )  =  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
) `  ( (
x ( 2nd `  F
) y ) `  f ) ) )
153134, 119, 152syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) `
 f )  =  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
) `  ( (
x ( 2nd `  F
) y ) `  f ) ) )
154151, 153eqtrd 2348 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( x ( 2nd `  ( G  o.func  F )
) y ) `  f )  =  ( ( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) ) `  (
( x ( 2nd `  F ) y ) `
 f ) ) )
155148, 149, 154oveq123d 5921 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( ( y ( 2nd `  ( G  o.func 
F ) ) z ) `  g ) ( <. ( ( 1st `  ( G  o.func  F )
) `  x ) ,  ( ( 1st `  ( G  o.func  F )
) `  y ) >. (comp `  E )
( ( 1st `  ( G  o.func 
F ) ) `  z ) ) ( ( x ( 2nd `  ( G  o.func  F )
) y ) `  f ) )  =  ( ( ( ( ( 1st `  F
) `  y )
( 2nd `  G
) ( ( 1st `  F ) `  z
) ) `  (
( y ( 2nd `  F ) z ) `
 g ) ) ( <. ( ( 1st `  G ) `  (
( 1st `  F
) `  x )
) ,  ( ( 1st `  G ) `
 ( ( 1st `  F ) `  y
) ) >. (comp `  E ) ( ( 1st `  G ) `
 ( ( 1st `  F ) `  z
) ) ) ( ( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) ) `  (
( x ( 2nd `  F ) y ) `
 f ) ) ) )
156139, 143, 1553eqtr4d 2358 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  C ) y )  /\  g  e.  ( y (  Hom  `  C
) z ) ) ) )  ->  (
( x ( 2nd `  ( G  o.func  F )
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( y ( 2nd `  ( G  o.func  F )
) z ) `  g ) ( <.
( ( 1st `  ( G  o.func 
F ) ) `  x ) ,  ( ( 1st `  ( G  o.func 
F ) ) `  y ) >. (comp `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  z ) ) ( ( x ( 2nd `  ( G  o.func  F )
) y ) `  f ) ) )
157156anassrs 629 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  ( y  e.  (
Base `  C )  /\  z  e.  ( Base `  C ) ) )  /\  ( f  e.  ( x (  Hom  `  C )
y )  /\  g  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( x ( 2nd `  ( G  o.func  F )
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( y ( 2nd `  ( G  o.func  F )
) z ) `  g ) ( <.
( ( 1st `  ( G  o.func 
F ) ) `  x ) ,  ( ( 1st `  ( G  o.func 
F ) ) `  y ) >. (comp `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  z ) ) ( ( x ( 2nd `  ( G  o.func  F )
) y ) `  f ) ) )
158157ralrimivva 2669 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  (
y  e.  ( Base `  C )  /\  z  e.  ( Base `  C
) ) )  ->  A. f  e.  (
x (  Hom  `  C
) y ) A. g  e.  ( y
(  Hom  `  C ) z ) ( ( x ( 2nd `  ( G  o.func 
F ) ) z ) `  ( g ( <. x ,  y
>. (comp `  C )
z ) f ) )  =  ( ( ( y ( 2nd `  ( G  o.func  F )
) z ) `  g ) ( <.
( ( 1st `  ( G  o.func 
F ) ) `  x ) ,  ( ( 1st `  ( G  o.func 
F ) ) `  y ) >. (comp `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  z ) ) ( ( x ( 2nd `  ( G  o.func  F )
) y ) `  f ) ) )
159158ralrimivva 2669 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  A. y  e.  ( Base `  C
) A. z  e.  ( Base `  C
) A. f  e.  ( x (  Hom  `  C ) y ) A. g  e.  ( y (  Hom  `  C
) z ) ( ( x ( 2nd `  ( G  o.func  F )
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( y ( 2nd `  ( G  o.func  F )
) z ) `  g ) ( <.
( ( 1st `  ( G  o.func 
F ) ) `  x ) ,  ( ( 1st `  ( G  o.func 
F ) ) `  y ) >. (comp `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  z ) ) ( ( x ( 2nd `  ( G  o.func  F )
) y ) `  f ) ) )
160111, 159jca 518 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
( x ( 2nd `  ( G  o.func  F )
) x ) `  ( ( Id `  C ) `  x
) )  =  ( ( Id `  E
) `  ( ( 1st `  ( G  o.func  F ) ) `  x ) )  /\  A. y  e.  ( Base `  C
) A. z  e.  ( Base `  C
) A. f  e.  ( x (  Hom  `  C ) y ) A. g  e.  ( y (  Hom  `  C
) z ) ( ( x ( 2nd `  ( G  o.func  F )
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( y ( 2nd `  ( G  o.func  F )
) z ) `  g ) ( <.
( ( 1st `  ( G  o.func 
F ) ) `  x ) ,  ( ( 1st `  ( G  o.func 
F ) ) `  y ) >. (comp `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  z ) ) ( ( x ( 2nd `  ( G  o.func  F )
) y ) `  f ) ) ) )
161160ralrimiva 2660 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  C )
( ( ( x ( 2nd `  ( G  o.func 
F ) ) x ) `  ( ( Id `  C ) `
 x ) )  =  ( ( Id
`  E ) `  ( ( 1st `  ( G  o.func 
F ) ) `  x ) )  /\  A. y  e.  ( Base `  C ) A. z  e.  ( Base `  C
) A. f  e.  ( x (  Hom  `  C ) y ) A. g  e.  ( y (  Hom  `  C
) z ) ( ( x ( 2nd `  ( G  o.func  F )
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( y ( 2nd `  ( G  o.func  F )
) z ) `  g ) ( <.
( ( 1st `  ( G  o.func 
F ) ) `  x ) ,  ( ( 1st `  ( G  o.func 
F ) ) `  y ) >. (comp `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  z ) ) ( ( x ( 2nd `  ( G  o.func  F )
) y ) `  f ) ) ) )
162 funcrcl 13786 . . . . . . 7  |-  ( G  e.  ( D  Func  E )  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
1633, 162syl 15 . . . . . 6  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
164163simprd 449 . . . . 5  |-  ( ph  ->  E  e.  Cat )
1651, 17, 49, 43, 85, 91, 116, 127, 103, 164isfunc 13787 . . . 4  |-  ( ph  ->  ( ( 1st `  ( G  o.func 
F ) ) ( C  Func  E )
( 2nd `  ( G  o.func 
F ) )  <->  ( ( 1st `  ( G  o.func  F ) ) : ( Base `  C ) --> ( Base `  E )  /\  ( 2nd `  ( G  o.func  F ) )  e.  X_ z  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ( ( ( ( 1st `  ( G  o.func  F )
) `  ( 1st `  z ) ) (  Hom  `  E )
( ( 1st `  ( G  o.func 
F ) ) `  ( 2nd `  z ) ) )  ^m  (
(  Hom  `  C ) `
 z ) )  /\  A. x  e.  ( Base `  C
) ( ( ( x ( 2nd `  ( G  o.func 
F ) ) x ) `  ( ( Id `  C ) `
 x ) )  =  ( ( Id
`  E ) `  ( ( 1st `  ( G  o.func 
F ) ) `  x ) )  /\  A. y  e.  ( Base `  C ) A. z  e.  ( Base `  C
) A. f  e.  ( x (  Hom  `  C ) y ) A. g  e.  ( y (  Hom  `  C
) z ) ( ( x ( 2nd `  ( G  o.func  F )
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( y ( 2nd `  ( G  o.func  F )
) z ) `  g ) ( <.
( ( 1st `  ( G  o.func 
F ) ) `  x ) ,  ( ( 1st `  ( G  o.func 
F ) ) `  y ) >. (comp `  E ) ( ( 1st `  ( G  o.func 
F ) ) `  z ) ) ( ( x ( 2nd `  ( G  o.func  F )
) y ) `  f ) ) ) ) ) )
16629, 84, 161, 165mpbir3and 1135 . . 3  |-  ( ph  ->  ( 1st `  ( G  o.func 
F ) ) ( C  Func  E )
( 2nd `  ( G  o.func 
F ) ) )
167 df-br 4061 . . 3  |-  ( ( 1st `  ( G  o.func 
F ) ) ( C  Func  E )
( 2nd `  ( G  o.func 
F ) )  <->  <. ( 1st `  ( G  o.func  F )
) ,  ( 2nd `  ( G  o.func  F )
) >.  e.  ( C 
Func  E ) )
168166, 167sylib 188 . 2  |-  ( ph  -> 
<. ( 1st `  ( G  o.func 
F ) ) ,  ( 2nd `  ( G  o.func 
F ) ) >.  e.  ( C  Func  E
) )
16915, 168eqeltrd 2390 1  |-  ( ph  ->  ( G  o.func  F )  e.  ( C  Func  E
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577   <.cop 3677   class class class wbr 4060    X. cxp 4724    o. ccom 4730   Rel wrel 4731    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   1stc1st 6162   2ndc2nd 6163    ^m cmap 6815   X_cixp 6860   Basecbs 13195    Hom chom 13266  compcco 13267   Catccat 13615   Idccid 13616    Func cfunc 13777    o.func ccofu 13779
This theorem is referenced by:  cofuass  13812  cofull  13857  cofth  13858  catccatid  13983  1st2ndprf  14029  uncfcl  14058  uncf1  14059  uncf2  14060  yonedalem1  14095  yonedalem21  14096  yonedalem22  14101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-map 6817  df-ixp 6861  df-cat 13619  df-cid 13620  df-func 13781  df-cofu 13783
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