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Theorem cofu2 13760
Description: Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-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 ) )
cofu2nd.x  |-  ( ph  ->  X  e.  B )
cofu2nd.y  |-  ( ph  ->  Y  e.  B )
cofu2.h  |-  H  =  (  Hom  `  C
)
cofu2.y  |-  ( ph  ->  R  e.  ( X H Y ) )
Assertion
Ref Expression
cofu2  |-  ( ph  ->  ( ( X ( 2nd `  ( G  o.func 
F ) ) Y ) `  R )  =  ( ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) ) `  (
( X ( 2nd `  F ) Y ) `
 R ) ) )

Proof of Theorem cofu2
StepHypRef Expression
1 cofuval.b . . . 4  |-  B  =  ( Base `  C
)
2 cofuval.f . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
3 cofuval.g . . . 4  |-  ( ph  ->  G  e.  ( D 
Func  E ) )
4 cofu2nd.x . . . 4  |-  ( ph  ->  X  e.  B )
5 cofu2nd.y . . . 4  |-  ( ph  ->  Y  e.  B )
61, 2, 3, 4, 5cofu2nd 13759 . . 3  |-  ( ph  ->  ( X ( 2nd `  ( G  o.func  F )
) Y )  =  ( ( ( ( 1st `  F ) `
 X ) ( 2nd `  G ) ( ( 1st `  F
) `  Y )
)  o.  ( X ( 2nd `  F
) Y ) ) )
76fveq1d 5527 . 2  |-  ( ph  ->  ( ( X ( 2nd `  ( G  o.func 
F ) ) Y ) `  R )  =  ( ( ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) )  o.  ( X ( 2nd `  F
) Y ) ) `
 R ) )
8 cofu2.h . . . 4  |-  H  =  (  Hom  `  C
)
9 eqid 2283 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
10 relfunc 13736 . . . . 5  |-  Rel  ( C  Func  D )
11 1st2ndbr 6169 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1210, 2, 11sylancr 644 . . . 4  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
131, 8, 9, 12, 4, 5funcf2 13742 . . 3  |-  ( ph  ->  ( X ( 2nd `  F ) Y ) : ( X H Y ) --> ( ( ( 1st `  F
) `  X )
(  Hom  `  D ) ( ( 1st `  F
) `  Y )
) )
14 cofu2.y . . 3  |-  ( ph  ->  R  e.  ( X H Y ) )
15 fvco3 5596 . . 3  |-  ( ( ( X ( 2nd `  F ) Y ) : ( X H Y ) --> ( ( ( 1st `  F
) `  X )
(  Hom  `  D ) ( ( 1st `  F
) `  Y )
)  /\  R  e.  ( X H Y ) )  ->  ( (
( ( ( 1st `  F ) `  X
) ( 2nd `  G
) ( ( 1st `  F ) `  Y
) )  o.  ( X ( 2nd `  F
) Y ) ) `
 R )  =  ( ( ( ( 1st `  F ) `
 X ) ( 2nd `  G ) ( ( 1st `  F
) `  Y )
) `  ( ( X ( 2nd `  F
) Y ) `  R ) ) )
1613, 14, 15syl2anc 642 . 2  |-  ( ph  ->  ( ( ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) )  o.  ( X ( 2nd `  F
) Y ) ) `
 R )  =  ( ( ( ( 1st `  F ) `
 X ) ( 2nd `  G ) ( ( 1st `  F
) `  Y )
) `  ( ( X ( 2nd `  F
) Y ) `  R ) ) )
177, 16eqtrd 2315 1  |-  ( ph  ->  ( ( X ( 2nd `  ( G  o.func 
F ) ) Y ) `  R )  =  ( ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) ) `  (
( X ( 2nd `  F ) Y ) `
 R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   class class class wbr 4023    o. ccom 4693   Rel wrel 4694   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Hom chom 13219    Func cfunc 13728    o.func ccofu 13730
This theorem is referenced by:  cofucl  13762  1st2ndprf  13980  uncf2  14011  yonedalem22  14052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-ixp 6818  df-func 13732  df-cofu 13734
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