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Theorem cofu1 15114
Description: Value of the object 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 )
Assertion
Ref Expression
cofu1  |-  ( ph  ->  ( ( 1st `  ( G  o.func 
F ) ) `  X )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  X ) ) )

Proof of Theorem cofu1
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 ) )
41, 2, 3cofu1st 15113 . . 3  |-  ( ph  ->  ( 1st `  ( G  o.func 
F ) )  =  ( ( 1st `  G
)  o.  ( 1st `  F ) ) )
54fveq1d 5868 . 2  |-  ( ph  ->  ( ( 1st `  ( G  o.func 
F ) ) `  X )  =  ( ( ( 1st `  G
)  o.  ( 1st `  F ) ) `  X ) )
6 eqid 2467 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
7 relfunc 15092 . . . . 5  |-  Rel  ( C  Func  D )
8 1st2ndbr 6834 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
97, 2, 8sylancr 663 . . . 4  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
101, 6, 9funcf1 15096 . . 3  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
11 cofu2nd.x . . 3  |-  ( ph  ->  X  e.  B )
12 fvco3 5945 . . 3  |-  ( ( ( 1st `  F
) : B --> ( Base `  D )  /\  X  e.  B )  ->  (
( ( 1st `  G
)  o.  ( 1st `  F ) ) `  X )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  X ) ) )
1310, 11, 12syl2anc 661 . 2  |-  ( ph  ->  ( ( ( 1st `  G )  o.  ( 1st `  F ) ) `
 X )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  X ) ) )
145, 13eqtrd 2508 1  |-  ( ph  ->  ( ( 1st `  ( G  o.func 
F ) ) `  X )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   class class class wbr 4447    o. ccom 5003   Rel wrel 5004   -->wf 5584   ` cfv 5588  (class class class)co 6285   1stc1st 6783   2ndc2nd 6784   Basecbs 14493    Func cfunc 15084    o.func ccofu 15086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-map 7423  df-ixp 7471  df-func 15088  df-cofu 15090
This theorem is referenced by:  cofucl  15118  cofuass  15119  cofull  15164  cofth  15165  catciso  15295  1st2ndprf  15336  uncf1  15366  uncf2  15367  yonedalem21  15403  yonedalem22  15408
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