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Theorem cofu2 15102
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 15101 . . 3  |-  ( ph  ->  ( X ( 2nd `  ( G  o.func  F )
) Y )  =  ( ( ( ( 1st `  F ) `
 X ) ( 2nd `  G ) ( ( 1st `  F
) `  Y )
)  o.  ( X ( 2nd `  F
) Y ) ) )
76fveq1d 5859 . 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 2460 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
10 relfunc 15078 . . . . 5  |-  Rel  ( C  Func  D )
11 1st2ndbr 6823 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1210, 2, 11sylancr 663 . . . 4  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
131, 8, 9, 12, 4, 5funcf2 15084 . . 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 5935 . . 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 661 . 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 2501 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 1374    e. wcel 1762   class class class wbr 4440    o. ccom 4996   Rel wrel 4997   -->wf 5575   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   Basecbs 14479   Hom chom 14555    Func cfunc 15070    o.func ccofu 15072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412  df-ixp 7460  df-func 15074  df-cofu 15076
This theorem is referenced by:  cofucl  15104  1st2ndprf  15322  uncf2  15353  yonedalem22  15394
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