MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cofuval Structured version   Unicode version

Theorem cofuval 14081
Description: Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofuval.b  |-  B  =  ( Base `  C
)
cofuval.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
cofuval.g  |-  ( ph  ->  G  e.  ( D 
Func  E ) )
Assertion
Ref Expression
cofuval  |-  ( ph  ->  ( G  o.func  F )  =  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) >. )
Distinct variable groups:    x, y, B    x, F, y    x, G, y    ph, x, y
Allowed substitution hints:    C( x, y)    D( x, y)    E( x, y)

Proof of Theorem cofuval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cofu 14059 . . 3  |-  o.func  =  (
g  e.  _V , 
f  e.  _V  |->  <.
( ( 1st `  g
)  o.  ( 1st `  f ) ) ,  ( x  e.  dom  dom  ( 2nd `  f
) ,  y  e. 
dom  dom  ( 2nd `  f
)  |->  ( ( ( ( 1st `  f
) `  x )
( 2nd `  g
) ( ( 1st `  f ) `  y
) )  o.  (
x ( 2nd `  f
) y ) ) ) >. )
21a1i 11 . 2  |-  ( ph  ->  o.func  =  ( g  e. 
_V ,  f  e. 
_V  |->  <. ( ( 1st `  g )  o.  ( 1st `  f ) ) ,  ( x  e. 
dom  dom  ( 2nd `  f
) ,  y  e. 
dom  dom  ( 2nd `  f
)  |->  ( ( ( ( 1st `  f
) `  x )
( 2nd `  g
) ( ( 1st `  f ) `  y
) )  o.  (
x ( 2nd `  f
) y ) ) ) >. ) )
3 simprl 734 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
g  =  G )
43fveq2d 5734 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 1st `  g
)  =  ( 1st `  G ) )
5 simprr 735 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
f  =  F )
65fveq2d 5734 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
74, 6coeq12d 5039 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  g
)  o.  ( 1st `  f ) )  =  ( ( 1st `  G
)  o.  ( 1st `  F ) ) )
85fveq2d 5734 . . . . . . . 8  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
98dmeqd 5074 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  f
)  =  dom  ( 2nd `  F ) )
10 cofuval.b . . . . . . . . . 10  |-  B  =  ( Base `  C
)
11 relfunc 14061 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
12 cofuval.f . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
13 1st2ndbr 6398 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1411, 12, 13sylancr 646 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
1510, 14funcfn2 14068 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( B  X.  B ) )
16 fndm 5546 . . . . . . . . 9  |-  ( ( 2nd `  F )  Fn  ( B  X.  B )  ->  dom  ( 2nd `  F )  =  ( B  X.  B ) )
1715, 16syl 16 . . . . . . . 8  |-  ( ph  ->  dom  ( 2nd `  F
)  =  ( B  X.  B ) )
1817adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  F
)  =  ( B  X.  B ) )
199, 18eqtrd 2470 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  f
)  =  ( B  X.  B ) )
2019dmeqd 5074 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  dom  ( 2nd `  f
)  =  dom  ( B  X.  B ) )
21 dmxpid 5091 . . . . 5  |-  dom  ( B  X.  B )  =  B
2220, 21syl6eq 2486 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  dom  ( 2nd `  f
)  =  B )
233fveq2d 5734 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  g
)  =  ( 2nd `  G ) )
246fveq1d 5732 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  f
) `  x )  =  ( ( 1st `  F ) `  x
) )
256fveq1d 5732 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  f
) `  y )  =  ( ( 1st `  F ) `  y
) )
2623, 24, 25oveq123d 6104 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( ( 1st `  f ) `  x
) ( 2nd `  g
) ( ( 1st `  f ) `  y
) )  =  ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) ) )
278oveqd 6100 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( x ( 2nd `  f ) y )  =  ( x ( 2nd `  F ) y ) )
2826, 27coeq12d 5039 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( ( ( 1st `  f ) `
 x ) ( 2nd `  g ) ( ( 1st `  f
) `  y )
)  o.  ( x ( 2nd `  f
) y ) )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )
2922, 22, 28mpt2eq123dv 6138 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( x  e.  dom  dom  ( 2nd `  f
) ,  y  e. 
dom  dom  ( 2nd `  f
)  |->  ( ( ( ( 1st `  f
) `  x )
( 2nd `  g
) ( ( 1st `  f ) `  y
) )  o.  (
x ( 2nd `  f
) y ) ) )  =  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) )
307, 29opeq12d 3994 . 2  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  <. ( ( 1st `  g
)  o.  ( 1st `  f ) ) ,  ( x  e.  dom  dom  ( 2nd `  f
) ,  y  e. 
dom  dom  ( 2nd `  f
)  |->  ( ( ( ( 1st `  f
) `  x )
( 2nd `  g
) ( ( 1st `  f ) `  y
) )  o.  (
x ( 2nd `  f
) y ) ) ) >.  =  <. ( ( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) ) >. )
31 cofuval.g . . 3  |-  ( ph  ->  G  e.  ( D 
Func  E ) )
32 elex 2966 . . 3  |-  ( G  e.  ( D  Func  E )  ->  G  e.  _V )
3331, 32syl 16 . 2  |-  ( ph  ->  G  e.  _V )
34 elex 2966 . . 3  |-  ( F  e.  ( C  Func  D )  ->  F  e.  _V )
3512, 34syl 16 . 2  |-  ( ph  ->  F  e.  _V )
36 opex 4429 . . 3  |-  <. (
( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) ) >.  e.  _V
3736a1i 11 . 2  |-  ( ph  -> 
<. ( ( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) ) >.  e.  _V )
382, 30, 33, 35, 37ovmpt2d 6203 1  |-  ( ph  ->  ( G  o.func  F )  =  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   class class class wbr 4214    X. cxp 4878   dom cdm 4880    o. ccom 4884   Rel wrel 4885    Fn wfn 5451   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1stc1st 6349   2ndc2nd 6350   Basecbs 13471    Func cfunc 14053    o.func ccofu 14055
This theorem is referenced by:  cofu1st  14082  cofu2nd  14084  cofuval2  14086  cofucl  14087  cofuass  14088  cofulid  14089  cofurid  14090  prf1st  14303  prf2nd  14304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-map 7022  df-ixp 7066  df-func 14057  df-cofu 14059
  Copyright terms: Public domain W3C validator