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Theorem cofu2 14011
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 14010 . . 3  |-  ( ph  ->  ( X ( 2nd `  ( G  o.func  F )
) Y )  =  ( ( ( ( 1st `  F ) `
 X ) ( 2nd `  G ) ( ( 1st `  F
) `  Y )
)  o.  ( X ( 2nd `  F
) Y ) ) )
76fveq1d 5671 . 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 2388 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
10 relfunc 13987 . . . . 5  |-  Rel  ( C  Func  D )
11 1st2ndbr 6336 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1210, 2, 11sylancr 645 . . . 4  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
131, 8, 9, 12, 4, 5funcf2 13993 . . 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 5740 . . 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 643 . 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 2420 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 1649    e. wcel 1717   class class class wbr 4154    o. ccom 4823   Rel wrel 4824   -->wf 5391   ` cfv 5395  (class class class)co 6021   1stc1st 6287   2ndc2nd 6288   Basecbs 13397    Hom chom 13468    Func cfunc 13979    o.func ccofu 13981
This theorem is referenced by:  cofucl  14013  1st2ndprf  14231  uncf2  14262  yonedalem22  14303
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-map 6957  df-ixp 7001  df-func 13983  df-cofu 13985
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