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Theorem cofu1 15372
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 15371 . . 3  |-  ( ph  ->  ( 1st `  ( G  o.func 
F ) )  =  ( ( 1st `  G
)  o.  ( 1st `  F ) ) )
54fveq1d 5850 . 2  |-  ( ph  ->  ( ( 1st `  ( G  o.func 
F ) ) `  X )  =  ( ( ( 1st `  G
)  o.  ( 1st `  F ) ) `  X ) )
6 eqid 2454 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
7 relfunc 15350 . . . . 5  |-  Rel  ( C  Func  D )
8 1st2ndbr 6822 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
97, 2, 8sylancr 661 . . . 4  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
101, 6, 9funcf1 15354 . . 3  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
11 cofu2nd.x . . 3  |-  ( ph  ->  X  e.  B )
12 fvco3 5925 . . 3  |-  ( ( ( 1st `  F
) : B --> ( Base `  D )  /\  X  e.  B )  ->  (
( ( 1st `  G
)  o.  ( 1st `  F ) ) `  X )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  X ) ) )
1310, 11, 12syl2anc 659 . 2  |-  ( ph  ->  ( ( ( 1st `  G )  o.  ( 1st `  F ) ) `
 X )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  X ) ) )
145, 13eqtrd 2495 1  |-  ( ph  ->  ( ( 1st `  ( G  o.func 
F ) ) `  X )  =  ( ( 1st `  G
) `  ( ( 1st `  F ) `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   class class class wbr 4439    o. ccom 4992   Rel wrel 4993   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   Basecbs 14716    Func cfunc 15342    o.func ccofu 15344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-ixp 7463  df-func 15346  df-cofu 15348
This theorem is referenced by:  cofucl  15376  cofuass  15377  cofull  15422  cofth  15423  catciso  15585  1st2ndprf  15674  uncf1  15704  uncf2  15705  yonedalem21  15741  yonedalem22  15746
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