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Theorem cofu2 15499
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 15498 . . 3  |-  ( ph  ->  ( X ( 2nd `  ( G  o.func  F )
) Y )  =  ( ( ( ( 1st `  F ) `
 X ) ( 2nd `  G ) ( ( 1st `  F
) `  Y )
)  o.  ( X ( 2nd `  F
) Y ) ) )
76fveq1d 5851 . 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 2402 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
10 relfunc 15475 . . . . 5  |-  Rel  ( C  Func  D )
11 1st2ndbr 6833 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1210, 2, 11sylancr 661 . . . 4  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
131, 8, 9, 12, 4, 5funcf2 15481 . . 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 5926 . . 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 659 . 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 2443 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 setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   class class class wbr 4395    o. ccom 4827   Rel wrel 4828   -->wf 5565   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783   Basecbs 14841   Hom chom 14920    Func cfunc 15467    o.func ccofu 15469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-map 7459  df-ixp 7508  df-func 15471  df-cofu 15473
This theorem is referenced by:  cofucl  15501  1st2ndprf  15799  uncf2  15830  yonedalem22  15871
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