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Theorem yonedalem3 14306
Description: Lemma for yoneda 14309. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y  |-  Y  =  (Yon `  C )
yoneda.b  |-  B  =  ( Base `  C
)
yoneda.1  |-  .1.  =  ( Id `  C )
yoneda.o  |-  O  =  (oppCat `  C )
yoneda.s  |-  S  =  ( SetCat `  U )
yoneda.t  |-  T  =  ( SetCat `  V )
yoneda.q  |-  Q  =  ( O FuncCat  S )
yoneda.h  |-  H  =  (HomF
`  Q )
yoneda.r  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
yoneda.e  |-  E  =  ( O evalF  S )
yoneda.z  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
yoneda.c  |-  ( ph  ->  C  e.  Cat )
yoneda.w  |-  ( ph  ->  V  e.  W )
yoneda.u  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
yoneda.v  |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U
)  C_  V )
yoneda.m  |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) ) )
Assertion
Ref Expression
yonedalem3  |-  ( ph  ->  M  e.  ( Z ( ( Q  X.c  O
) Nat  T ) E ) )
Distinct variable groups:    f, a, x,  .1.    C, a, f, x    E, a, f    B, a, f, x    O, a, f, x    S, a, f, x    Q, a, f, x    T, f    ph, a, f, x    Y, a, f, x    Z, a, f, x
Allowed substitution hints:    R( x, f, a)    T( x, a)    U( x, f, a)    E( x)    H( x, f, a)    M( x, f, a)    V( x, f, a)    W( x, f, a)

Proof of Theorem yonedalem3
Dummy variables  g 
y  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.m . . . . 5  |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) ) )
2 ovex 6047 . . . . . 6  |-  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  e.  _V
32mptex 5907 . . . . 5  |-  ( a  e.  ( ( ( 1st `  Y ) `
 x ) ( O Nat  S ) f )  |->  ( ( a `
 x ) `  (  .1.  `  x )
) )  e.  _V
41, 3fnmpt2i 6361 . . . 4  |-  M  Fn  ( ( O  Func  S )  X.  B )
54a1i 11 . . 3  |-  ( ph  ->  M  Fn  ( ( O  Func  S )  X.  B ) )
6 yoneda.y . . . . . . . 8  |-  Y  =  (Yon `  C )
7 yoneda.b . . . . . . . 8  |-  B  =  ( Base `  C
)
8 yoneda.1 . . . . . . . 8  |-  .1.  =  ( Id `  C )
9 yoneda.o . . . . . . . 8  |-  O  =  (oppCat `  C )
10 yoneda.s . . . . . . . 8  |-  S  =  ( SetCat `  U )
11 yoneda.t . . . . . . . 8  |-  T  =  ( SetCat `  V )
12 yoneda.q . . . . . . . 8  |-  Q  =  ( O FuncCat  S )
13 yoneda.h . . . . . . . 8  |-  H  =  (HomF
`  Q )
14 yoneda.r . . . . . . . 8  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
15 yoneda.e . . . . . . . 8  |-  E  =  ( O evalF  S )
16 yoneda.z . . . . . . . 8  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
17 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
1817adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  ->  C  e.  Cat )
19 yoneda.w . . . . . . . . 9  |-  ( ph  ->  V  e.  W )
2019adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  ->  V  e.  W )
21 yoneda.u . . . . . . . . 9  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
2221adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  ->  ran  (  Homf 
`  C )  C_  U )
23 yoneda.v . . . . . . . . 9  |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U
)  C_  V )
2423adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( ran  (  Homf  `  Q
)  u.  U ) 
C_  V )
25 simprl 733 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
g  e.  ( O 
Func  S ) )
26 simprr 734 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
y  e.  B )
276, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 1yonedalem3a 14300 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( ( g M y )  =  ( a  e.  ( ( ( 1st `  Y
) `  y )
( O Nat  S ) g )  |->  ( ( a `  y ) `
 (  .1.  `  y ) ) )  /\  ( g M y ) : ( g ( 1st `  Z
) y ) --> ( g ( 1st `  E
) y ) ) )
2827simprd 450 . . . . . 6  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g M y ) : ( g ( 1st `  Z
) y ) --> ( g ( 1st `  E
) y ) )
29 eqid 2389 . . . . . . 7  |-  (  Hom  `  T )  =  (  Hom  `  T )
30 eqid 2389 . . . . . . . . . . 11  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
3112fucbas 14086 . . . . . . . . . . 11  |-  ( O 
Func  S )  =  (
Base `  Q )
329, 7oppcbas 13873 . . . . . . . . . . 11  |-  B  =  ( Base `  O
)
3330, 31, 32xpcbas 14204 . . . . . . . . . 10  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
34 eqid 2389 . . . . . . . . . 10  |-  ( Base `  T )  =  (
Base `  T )
35 relfunc 13988 . . . . . . . . . . 11  |-  Rel  (
( Q  X.c  O ) 
Func  T )
366, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23yonedalem1 14298 . . . . . . . . . . . 12  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )
3736simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
38 1st2ndbr 6337 . . . . . . . . . . 11  |-  ( ( Rel  ( ( Q  X.c  O )  Func  T
)  /\  Z  e.  ( ( Q  X.c  O
)  Func  T )
)  ->  ( 1st `  Z ) ( ( Q  X.c  O )  Func  T
) ( 2nd `  Z
) )
3935, 37, 38sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  Z
) ( ( Q  X.c  O )  Func  T
) ( 2nd `  Z
) )
4033, 34, 39funcf1 13992 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  Z
) : ( ( O  Func  S )  X.  B ) --> ( Base `  T ) )
4140fovrnda 6158 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  Z ) y )  e.  ( Base `  T
) )
4211, 20setcbas 14162 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  ->  V  =  ( Base `  T ) )
4341, 42eleqtrrd 2466 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  Z ) y )  e.  V )
4436simprd 450 . . . . . . . . . . 11  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
45 1st2ndbr 6337 . . . . . . . . . . 11  |-  ( ( Rel  ( ( Q  X.c  O )  Func  T
)  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
)  ->  ( 1st `  E ) ( ( Q  X.c  O )  Func  T
) ( 2nd `  E
) )
4635, 44, 45sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  E
) ( ( Q  X.c  O )  Func  T
) ( 2nd `  E
) )
4733, 34, 46funcf1 13992 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  E
) : ( ( O  Func  S )  X.  B ) --> ( Base `  T ) )
4847fovrnda 6158 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  E ) y )  e.  ( Base `  T
) )
4948, 42eleqtrrd 2466 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  E ) y )  e.  V )
5011, 20, 29, 43, 49elsetchom 14165 . . . . . 6  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( ( g M y )  e.  ( ( g ( 1st `  Z ) y ) (  Hom  `  T
) ( g ( 1st `  E ) y ) )  <->  ( g M y ) : ( g ( 1st `  Z ) y ) --> ( g ( 1st `  E ) y ) ) )
5128, 50mpbird 224 . . . . 5  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g M y )  e.  ( ( g ( 1st `  Z
) y ) (  Hom  `  T )
( g ( 1st `  E ) y ) ) )
5251ralrimivva 2743 . . . 4  |-  ( ph  ->  A. g  e.  ( O  Func  S ) A. y  e.  B  ( g M y )  e.  ( ( g ( 1st `  Z
) y ) (  Hom  `  T )
( g ( 1st `  E ) y ) ) )
53 fveq2 5670 . . . . . . 7  |-  ( z  =  <. g ,  y
>.  ->  ( M `  z )  =  ( M `  <. g ,  y >. )
)
54 df-ov 6025 . . . . . . 7  |-  ( g M y )  =  ( M `  <. g ,  y >. )
5553, 54syl6eqr 2439 . . . . . 6  |-  ( z  =  <. g ,  y
>.  ->  ( M `  z )  =  ( g M y ) )
56 fveq2 5670 . . . . . . . 8  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  Z ) `  z
)  =  ( ( 1st `  Z ) `
 <. g ,  y
>. ) )
57 df-ov 6025 . . . . . . . 8  |-  ( g ( 1st `  Z
) y )  =  ( ( 1st `  Z
) `  <. g ,  y >. )
5856, 57syl6eqr 2439 . . . . . . 7  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  Z ) `  z
)  =  ( g ( 1st `  Z
) y ) )
59 fveq2 5670 . . . . . . . 8  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  E ) `  z
)  =  ( ( 1st `  E ) `
 <. g ,  y
>. ) )
60 df-ov 6025 . . . . . . . 8  |-  ( g ( 1st `  E
) y )  =  ( ( 1st `  E
) `  <. g ,  y >. )
6159, 60syl6eqr 2439 . . . . . . 7  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  E ) `  z
)  =  ( g ( 1st `  E
) y ) )
6258, 61oveq12d 6040 . . . . . 6  |-  ( z  =  <. g ,  y
>.  ->  ( ( ( 1st `  Z ) `
 z ) (  Hom  `  T )
( ( 1st `  E
) `  z )
)  =  ( ( g ( 1st `  Z
) y ) (  Hom  `  T )
( g ( 1st `  E ) y ) ) )
6355, 62eleq12d 2457 . . . . 5  |-  ( z  =  <. g ,  y
>.  ->  ( ( M `
 z )  e.  ( ( ( 1st `  Z ) `  z
) (  Hom  `  T
) ( ( 1st `  E ) `  z
) )  <->  ( g M y )  e.  ( ( g ( 1st `  Z ) y ) (  Hom  `  T ) ( g ( 1st `  E
) y ) ) ) )
6463ralxp 4958 . . . 4  |-  ( A. z  e.  ( ( O  Func  S )  X.  B ) ( M `
 z )  e.  ( ( ( 1st `  Z ) `  z
) (  Hom  `  T
) ( ( 1st `  E ) `  z
) )  <->  A. g  e.  ( O  Func  S
) A. y  e.  B  ( g M y )  e.  ( ( g ( 1st `  Z ) y ) (  Hom  `  T
) ( g ( 1st `  E ) y ) ) )
6552, 64sylibr 204 . . 3  |-  ( ph  ->  A. z  e.  ( ( O  Func  S
)  X.  B ) ( M `  z
)  e.  ( ( ( 1st `  Z
) `  z )
(  Hom  `  T ) ( ( 1st `  E
) `  z )
) )
66 ovex 6047 . . . . . 6  |-  ( O 
Func  S )  e.  _V
67 fvex 5684 . . . . . . 7  |-  ( Base `  C )  e.  _V
687, 67eqeltri 2459 . . . . . 6  |-  B  e. 
_V
6966, 68mpt2ex 6366 . . . . 5  |-  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x
) ( O Nat  S
) f )  |->  ( ( a `  x
) `  (  .1.  `  x ) ) ) )  e.  _V
701, 69eqeltri 2459 . . . 4  |-  M  e. 
_V
7170elixp 7007 . . 3  |-  ( M  e.  X_ z  e.  ( ( O  Func  S
)  X.  B ) ( ( ( 1st `  Z ) `  z
) (  Hom  `  T
) ( ( 1st `  E ) `  z
) )  <->  ( M  Fn  ( ( O  Func  S )  X.  B )  /\  A. z  e.  ( ( O  Func  S )  X.  B ) ( M `  z
)  e.  ( ( ( 1st `  Z
) `  z )
(  Hom  `  T ) ( ( 1st `  E
) `  z )
) ) )
725, 65, 71sylanbrc 646 . 2  |-  ( ph  ->  M  e.  X_ z  e.  ( ( O  Func  S )  X.  B ) ( ( ( 1st `  Z ) `  z
) (  Hom  `  T
) ( ( 1st `  E ) `  z
) ) )
7317adantr 452 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  C  e.  Cat )
7419adantr 452 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  V  e.  W )
7521adantr 452 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  ran  (  Homf 
`  C )  C_  U )
7623adantr 452 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ran  (  Homf  `  Q
)  u.  U ) 
C_  V )
77 simpr1 963 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
z  e.  ( ( O  Func  S )  X.  B ) )
78 xp1st 6317 . . . . . 6  |-  ( z  e.  ( ( O 
Func  S )  X.  B
)  ->  ( 1st `  z )  e.  ( O  Func  S )
)
7977, 78syl 16 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( 1st `  z
)  e.  ( O 
Func  S ) )
80 xp2nd 6318 . . . . . 6  |-  ( z  e.  ( ( O 
Func  S )  X.  B
)  ->  ( 2nd `  z )  e.  B
)
8177, 80syl 16 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( 2nd `  z
)  e.  B )
82 simpr2 964 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  w  e.  ( ( O  Func  S )  X.  B ) )
83 xp1st 6317 . . . . . 6  |-  ( w  e.  ( ( O 
Func  S )  X.  B
)  ->  ( 1st `  w )  e.  ( O  Func  S )
)
8482, 83syl 16 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( 1st `  w
)  e.  ( O 
Func  S ) )
85 xp2nd 6318 . . . . . 6  |-  ( w  e.  ( ( O 
Func  S )  X.  B
)  ->  ( 2nd `  w )  e.  B
)
8682, 85syl 16 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( 2nd `  w
)  e.  B )
87 simpr3 965 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) )
88 eqid 2389 . . . . . . . . . 10  |-  ( O Nat 
S )  =  ( O Nat  S )
8912, 88fuchom 14087 . . . . . . . . 9  |-  ( O Nat 
S )  =  (  Hom  `  Q )
90 eqid 2389 . . . . . . . . 9  |-  (  Hom  `  O )  =  (  Hom  `  O )
91 eqid 2389 . . . . . . . . 9  |-  (  Hom  `  ( Q  X.c  O ) )  =  (  Hom  `  ( Q  X.c  O ) )
9230, 33, 89, 90, 91, 77, 82xpchom 14206 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( z (  Hom  `  ( Q  X.c  O ) ) w )  =  ( ( ( 1st `  z ) ( O Nat 
S ) ( 1st `  w ) )  X.  ( ( 2nd `  z
) (  Hom  `  O
) ( 2nd `  w
) ) ) )
93 eqid 2389 . . . . . . . . . 10  |-  (  Hom  `  C )  =  (  Hom  `  C )
9493, 9oppchom 13870 . . . . . . . . 9  |-  ( ( 2nd `  z ) (  Hom  `  O
) ( 2nd `  w
) )  =  ( ( 2nd `  w
) (  Hom  `  C
) ( 2nd `  z
) )
9594xpeq2i 4841 . . . . . . . 8  |-  ( ( ( 1st `  z
) ( O Nat  S
) ( 1st `  w
) )  X.  (
( 2nd `  z
) (  Hom  `  O
) ( 2nd `  w
) ) )  =  ( ( ( 1st `  z ) ( O Nat 
S ) ( 1st `  w ) )  X.  ( ( 2nd `  w
) (  Hom  `  C
) ( 2nd `  z
) ) )
9692, 95syl6eq 2437 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( z (  Hom  `  ( Q  X.c  O ) ) w )  =  ( ( ( 1st `  z ) ( O Nat 
S ) ( 1st `  w ) )  X.  ( ( 2nd `  w
) (  Hom  `  C
) ( 2nd `  z
) ) ) )
9787, 96eleqtrd 2465 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
g  e.  ( ( ( 1st `  z
) ( O Nat  S
) ( 1st `  w
) )  X.  (
( 2nd `  w
) (  Hom  `  C
) ( 2nd `  z
) ) ) )
98 xp1st 6317 . . . . . 6  |-  ( g  e.  ( ( ( 1st `  z ) ( O Nat  S ) ( 1st `  w
) )  X.  (
( 2nd `  w
) (  Hom  `  C
) ( 2nd `  z
) ) )  -> 
( 1st `  g
)  e.  ( ( 1st `  z ) ( O Nat  S ) ( 1st `  w
) ) )
9997, 98syl 16 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( 1st `  g
)  e.  ( ( 1st `  z ) ( O Nat  S ) ( 1st `  w
) ) )
100 xp2nd 6318 . . . . . 6  |-  ( g  e.  ( ( ( 1st `  z ) ( O Nat  S ) ( 1st `  w
) )  X.  (
( 2nd `  w
) (  Hom  `  C
) ( 2nd `  z
) ) )  -> 
( 2nd `  g
)  e.  ( ( 2nd `  w ) (  Hom  `  C
) ( 2nd `  z
) ) )
10197, 100syl 16 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( 2nd `  g
)  e.  ( ( 2nd `  w ) (  Hom  `  C
) ( 2nd `  z
) ) )
1026, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 73, 74, 75, 76, 79, 81, 84, 86, 99, 101, 1yonedalem3b 14305 . . . 4  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( ( 1st `  w ) M ( 2nd `  w ) ) ( <. (
( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  w
) ( 1st `  Z
) ( 2nd `  w
) ) >. (comp `  T ) ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) ) ) ( ( 1st `  g
) ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. ( 2nd `  Z
) <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) ) )  =  ( ( ( 1st `  g ) ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  E ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) ) ( <.
( ( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  z
) ( 1st `  E
) ( 2nd `  z
) ) >. (comp `  T ) ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) ) ) ( ( 1st `  z
) M ( 2nd `  z ) ) ) )
103 1st2nd2 6327 . . . . . . . . . 10  |-  ( z  e.  ( ( O 
Func  S )  X.  B
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
10477, 103syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
105104fveq2d 5674 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  Z
) `  z )  =  ( ( 1st `  Z ) `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
106 df-ov 6025 . . . . . . . 8  |-  ( ( 1st `  z ) ( 1st `  Z
) ( 2nd `  z
) )  =  ( ( 1st `  Z
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
107105, 106syl6eqr 2439 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  Z
) `  z )  =  ( ( 1st `  z ) ( 1st `  Z ) ( 2nd `  z ) ) )
108 1st2nd2 6327 . . . . . . . . . 10  |-  ( w  e.  ( ( O 
Func  S )  X.  B
)  ->  w  =  <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
10982, 108syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  w  =  <. ( 1st `  w ) ,  ( 2nd `  w )
>. )
110109fveq2d 5674 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  Z
) `  w )  =  ( ( 1st `  Z ) `  <. ( 1st `  w ) ,  ( 2nd `  w
) >. ) )
111 df-ov 6025 . . . . . . . 8  |-  ( ( 1st `  w ) ( 1st `  Z
) ( 2nd `  w
) )  =  ( ( 1st `  Z
) `  <. ( 1st `  w ) ,  ( 2nd `  w )
>. )
112110, 111syl6eqr 2439 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  Z
) `  w )  =  ( ( 1st `  w ) ( 1st `  Z ) ( 2nd `  w ) ) )
113107, 112opeq12d 3936 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  <. ( ( 1st `  Z
) `  z ) ,  ( ( 1st `  Z ) `  w
) >.  =  <. (
( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  w
) ( 1st `  Z
) ( 2nd `  w
) ) >. )
114109fveq2d 5674 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  E
) `  w )  =  ( ( 1st `  E ) `  <. ( 1st `  w ) ,  ( 2nd `  w
) >. ) )
115 df-ov 6025 . . . . . . 7  |-  ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) )  =  ( ( 1st `  E
) `  <. ( 1st `  w ) ,  ( 2nd `  w )
>. )
116114, 115syl6eqr 2439 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  E
) `  w )  =  ( ( 1st `  w ) ( 1st `  E ) ( 2nd `  w ) ) )
117113, 116oveq12d 6040 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( <. ( ( 1st `  Z ) `  z
) ,  ( ( 1st `  Z ) `
 w ) >.
(comp `  T )
( ( 1st `  E
) `  w )
)  =  ( <.
( ( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  w
) ( 1st `  Z
) ( 2nd `  w
) ) >. (comp `  T ) ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) ) ) )
118109fveq2d 5674 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( M `  w
)  =  ( M `
 <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
)
119 df-ov 6025 . . . . . 6  |-  ( ( 1st `  w ) M ( 2nd `  w
) )  =  ( M `  <. ( 1st `  w ) ,  ( 2nd `  w
) >. )
120118, 119syl6eqr 2439 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( M `  w
)  =  ( ( 1st `  w ) M ( 2nd `  w
) ) )
121104, 109oveq12d 6040 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( z ( 2nd `  Z ) w )  =  ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. ( 2nd `  Z
) <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
)
122 1st2nd2 6327 . . . . . . . 8  |-  ( g  e.  ( ( ( 1st `  z ) ( O Nat  S ) ( 1st `  w
) )  X.  (
( 2nd `  w
) (  Hom  `  C
) ( 2nd `  z
) ) )  -> 
g  =  <. ( 1st `  g ) ,  ( 2nd `  g
) >. )
12397, 122syl 16 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
g  =  <. ( 1st `  g ) ,  ( 2nd `  g
) >. )
124121, 123fveq12d 5676 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( z ( 2nd `  Z ) w ) `  g
)  =  ( (
<. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  Z ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. ) `  <. ( 1st `  g
) ,  ( 2nd `  g ) >. )
)
125 df-ov 6025 . . . . . 6  |-  ( ( 1st `  g ) ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  Z ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) )  =  ( ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  Z ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. ) `  <. ( 1st `  g
) ,  ( 2nd `  g ) >. )
126124, 125syl6eqr 2439 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( z ( 2nd `  Z ) w ) `  g
)  =  ( ( 1st `  g ) ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  Z ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) ) )
127117, 120, 126oveq123d 6043 . . . 4  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( M `  w ) ( <.
( ( 1st `  Z
) `  z ) ,  ( ( 1st `  Z ) `  w
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( ( z ( 2nd `  Z
) w ) `  g ) )  =  ( ( ( 1st `  w ) M ( 2nd `  w ) ) ( <. (
( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  w
) ( 1st `  Z
) ( 2nd `  w
) ) >. (comp `  T ) ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) ) ) ( ( 1st `  g
) ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. ( 2nd `  Z
) <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) ) ) )
128104fveq2d 5674 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  E
) `  z )  =  ( ( 1st `  E ) `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
129 df-ov 6025 . . . . . . . 8  |-  ( ( 1st `  z ) ( 1st `  E
) ( 2nd `  z
) )  =  ( ( 1st `  E
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
130128, 129syl6eqr 2439 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  E
) `  z )  =  ( ( 1st `  z ) ( 1st `  E ) ( 2nd `  z ) ) )
131107, 130opeq12d 3936 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  <. ( ( 1st `  Z
) `  z ) ,  ( ( 1st `  E ) `  z
) >.  =  <. (
( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  z
) ( 1st `  E
) ( 2nd `  z
) ) >. )
132131, 116oveq12d 6040 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( <. ( ( 1st `  Z ) `  z
) ,  ( ( 1st `  E ) `
 z ) >.
(comp `  T )
( ( 1st `  E
) `  w )
)  =  ( <.
( ( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  z
) ( 1st `  E
) ( 2nd `  z
) ) >. (comp `  T ) ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) ) ) )
133104, 109oveq12d 6040 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( z ( 2nd `  E ) w )  =  ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. ( 2nd `  E
) <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
)
134133, 123fveq12d 5676 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( z ( 2nd `  E ) w ) `  g
)  =  ( (
<. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  E ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. ) `  <. ( 1st `  g
) ,  ( 2nd `  g ) >. )
)
135 df-ov 6025 . . . . . 6  |-  ( ( 1st `  g ) ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  E ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) )  =  ( ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  E ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. ) `  <. ( 1st `  g
) ,  ( 2nd `  g ) >. )
136134, 135syl6eqr 2439 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( z ( 2nd `  E ) w ) `  g
)  =  ( ( 1st `  g ) ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  E ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) ) )
137104fveq2d 5674 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( M `  z
)  =  ( M `
 <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
)
138 df-ov 6025 . . . . . 6  |-  ( ( 1st `  z ) M ( 2nd `  z
) )  =  ( M `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
139137, 138syl6eqr 2439 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( M `  z
)  =  ( ( 1st `  z ) M ( 2nd `  z
) ) )
140132, 136, 139oveq123d 6043 . . . 4  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( ( z ( 2nd `  E
) w ) `  g ) ( <.
( ( 1st `  Z
) `  z ) ,  ( ( 1st `  E ) `  z
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( M `
 z ) )  =  ( ( ( 1st `  g ) ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  E ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) ) ( <.
( ( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  z
) ( 1st `  E
) ( 2nd `  z
) ) >. (comp `  T ) ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) ) ) ( ( 1st `  z
) M ( 2nd `  z ) ) ) )
141102, 127, 1403eqtr4d 2431 . . 3  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( M `  w ) ( <.
( ( 1st `  Z
) `  z ) ,  ( ( 1st `  Z ) `  w
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( ( z ( 2nd `  Z
) w ) `  g ) )  =  ( ( ( z ( 2nd `  E
) w ) `  g ) ( <.
( ( 1st `  Z
) `  z ) ,  ( ( 1st `  E ) `  z
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( M `
 z ) ) )
142141ralrimivvva 2744 . 2  |-  ( ph  ->  A. z  e.  ( ( O  Func  S
)  X.  B ) A. w  e.  ( ( O  Func  S
)  X.  B ) A. g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ( ( M `
 w ) (
<. ( ( 1st `  Z
) `  z ) ,  ( ( 1st `  Z ) `  w
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( ( z ( 2nd `  Z
) w ) `  g ) )  =  ( ( ( z ( 2nd `  E
) w ) `  g ) ( <.
( ( 1st `  Z
) `  z ) ,  ( ( 1st `  E ) `  z
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( M `
 z ) ) )
143 eqid 2389 . . 3  |-  ( ( Q  X.c  O ) Nat  T )  =  ( ( Q  X.c  O ) Nat  T )
144 eqid 2389 . . 3  |-  (comp `  T )  =  (comp `  T )
145143, 33, 91, 29, 144, 37, 44isnat2 14074 . 2  |-  ( ph  ->  ( M  e.  ( Z ( ( Q  X.c  O ) Nat  T ) E )  <->  ( M  e.  X_ z  e.  ( ( O  Func  S
)  X.  B ) ( ( ( 1st `  Z ) `  z
) (  Hom  `  T
) ( ( 1st `  E ) `  z
) )  /\  A. z  e.  ( ( O  Func  S )  X.  B ) A. w  e.  ( ( O  Func  S )  X.  B ) A. g  e.  ( z (  Hom  `  ( Q  X.c  O ) ) w ) ( ( M `
 w ) (
<. ( ( 1st `  Z
) `  z ) ,  ( ( 1st `  Z ) `  w
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( ( z ( 2nd `  Z
) w ) `  g ) )  =  ( ( ( z ( 2nd `  E
) w ) `  g ) ( <.
( ( 1st `  Z
) `  z ) ,  ( ( 1st `  E ) `  z
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( M `
 z ) ) ) ) )
14672, 142, 145mpbir2and 889 1  |-  ( ph  ->  M  e.  ( Z ( ( Q  X.c  O
) Nat  T ) E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   _Vcvv 2901    u. cun 3263    C_ wss 3265   <.cop 3762   class class class wbr 4155    e. cmpt 4209    X. cxp 4818   ran crn 4821   Rel wrel 4825    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022    e. cmpt2 6024   1stc1st 6288   2ndc2nd 6289  tpos ctpos 6416   X_cixp 7001   Basecbs 13398    Hom chom 13469  compcco 13470   Catccat 13818   Idccid 13819    Homf chomf 13820  oppCatcoppc 13866    Func cfunc 13980    o.func ccofu 13982   Nat cnat 14067   FuncCat cfuc 14068   SetCatcsetc 14159    X.c cxpc 14194    1stF c1stf 14195    2ndF c2ndf 14196   ⟨,⟩F cprf 14197   evalF cevlf 14235  HomFchof 14274  Yoncyon 14275
This theorem is referenced by:  yonedainv  14307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-tpos 6417  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-fz 10978  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-hom 13482  df-cco 13483  df-cat 13822  df-cid 13823  df-homf 13824  df-comf 13825  df-oppc 13867  df-ssc 13939  df-resc 13940  df-subc 13941  df-func 13984  df-cofu 13986  df-nat 14069  df-fuc 14070  df-setc 14160  df-xpc 14198  df-1stf 14199  df-2ndf 14200  df-prf 14201  df-evlf 14239  df-curf 14240  df-hof 14276  df-yon 14277
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