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Theorem funcf2 15112
Description: The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
funcixp.b  |-  B  =  ( Base `  D
)
funcixp.h  |-  H  =  ( Hom  `  D
)
funcixp.j  |-  J  =  ( Hom  `  E
)
funcixp.f  |-  ( ph  ->  F ( D  Func  E ) G )
funcf2.x  |-  ( ph  ->  X  e.  B )
funcf2.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
funcf2  |-  ( ph  ->  ( X G Y ) : ( X H Y ) --> ( ( F `  X
) J ( F `
 Y ) ) )

Proof of Theorem funcf2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-ov 6298 . . . 4  |-  ( X G Y )  =  ( G `  <. X ,  Y >. )
2 funcixp.b . . . . . 6  |-  B  =  ( Base `  D
)
3 funcixp.h . . . . . 6  |-  H  =  ( Hom  `  D
)
4 funcixp.j . . . . . 6  |-  J  =  ( Hom  `  E
)
5 funcixp.f . . . . . 6  |-  ( ph  ->  F ( D  Func  E ) G )
62, 3, 4, 5funcixp 15111 . . . . 5  |-  ( ph  ->  G  e.  X_ z  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  z ) ) J ( F `  ( 2nd `  z ) ) )  ^m  ( H `
 z ) ) )
7 funcf2.x . . . . . 6  |-  ( ph  ->  X  e.  B )
8 funcf2.y . . . . . 6  |-  ( ph  ->  Y  e.  B )
9 opelxpi 5037 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
107, 8, 9syl2anc 661 . . . . 5  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
11 fveq2 5872 . . . . . . . . 9  |-  ( z  =  <. X ,  Y >.  ->  ( 1st `  z
)  =  ( 1st `  <. X ,  Y >. ) )
1211fveq2d 5876 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( F `  ( 1st `  z ) )  =  ( F `
 ( 1st `  <. X ,  Y >. )
) )
13 fveq2 5872 . . . . . . . . 9  |-  ( z  =  <. X ,  Y >.  ->  ( 2nd `  z
)  =  ( 2nd `  <. X ,  Y >. ) )
1413fveq2d 5876 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( F `  ( 2nd `  z ) )  =  ( F `
 ( 2nd `  <. X ,  Y >. )
) )
1512, 14oveq12d 6313 . . . . . . 7  |-  ( z  =  <. X ,  Y >.  ->  ( ( F `
 ( 1st `  z
) ) J ( F `  ( 2nd `  z ) ) )  =  ( ( F `
 ( 1st `  <. X ,  Y >. )
) J ( F `
 ( 2nd `  <. X ,  Y >. )
) ) )
16 fveq2 5872 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
17 df-ov 6298 . . . . . . . 8  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1816, 17syl6eqr 2526 . . . . . . 7  |-  ( z  =  <. X ,  Y >.  ->  ( H `  z )  =  ( X H Y ) )
1915, 18oveq12d 6313 . . . . . 6  |-  ( z  =  <. X ,  Y >.  ->  ( ( ( F `  ( 1st `  z ) ) J ( F `  ( 2nd `  z ) ) )  ^m  ( H `
 z ) )  =  ( ( ( F `  ( 1st `  <. X ,  Y >. ) ) J ( F `  ( 2nd `  <. X ,  Y >. ) ) )  ^m  ( X H Y ) ) )
2019fvixp 7486 . . . . 5  |-  ( ( G  e.  X_ z  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  z ) ) J ( F `  ( 2nd `  z ) ) )  ^m  ( H `
 z ) )  /\  <. X ,  Y >.  e.  ( B  X.  B ) )  -> 
( G `  <. X ,  Y >. )  e.  ( ( ( F `
 ( 1st `  <. X ,  Y >. )
) J ( F `
 ( 2nd `  <. X ,  Y >. )
) )  ^m  ( X H Y ) ) )
216, 10, 20syl2anc 661 . . . 4  |-  ( ph  ->  ( G `  <. X ,  Y >. )  e.  ( ( ( F `
 ( 1st `  <. X ,  Y >. )
) J ( F `
 ( 2nd `  <. X ,  Y >. )
) )  ^m  ( X H Y ) ) )
221, 21syl5eqel 2559 . . 3  |-  ( ph  ->  ( X G Y )  e.  ( ( ( F `  ( 1st `  <. X ,  Y >. ) ) J ( F `  ( 2nd `  <. X ,  Y >. ) ) )  ^m  ( X H Y ) ) )
23 op1stg 6807 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
2423fveq2d 5876 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( 1st `  <. X ,  Y >. ) )  =  ( F `  X ) )
25 op2ndg 6808 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2625fveq2d 5876 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( 2nd `  <. X ,  Y >. ) )  =  ( F `  Y ) )
2724, 26oveq12d 6313 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( F `  ( 1st `  <. X ,  Y >. ) ) J ( F `  ( 2nd `  <. X ,  Y >. ) ) )  =  ( ( F `  X ) J ( F `  Y ) ) )
287, 8, 27syl2anc 661 . . . 4  |-  ( ph  ->  ( ( F `  ( 1st `  <. X ,  Y >. ) ) J ( F `  ( 2nd `  <. X ,  Y >. ) ) )  =  ( ( F `  X ) J ( F `  Y ) ) )
2928oveq1d 6310 . . 3  |-  ( ph  ->  ( ( ( F `
 ( 1st `  <. X ,  Y >. )
) J ( F `
 ( 2nd `  <. X ,  Y >. )
) )  ^m  ( X H Y ) )  =  ( ( ( F `  X ) J ( F `  Y ) )  ^m  ( X H Y ) ) )
3022, 29eleqtrd 2557 . 2  |-  ( ph  ->  ( X G Y )  e.  ( ( ( F `  X
) J ( F `
 Y ) )  ^m  ( X H Y ) ) )
31 elmapi 7452 . 2  |-  ( ( X G Y )  e.  ( ( ( F `  X ) J ( F `  Y ) )  ^m  ( X H Y ) )  ->  ( X G Y ) : ( X H Y ) --> ( ( F `  X ) J ( F `  Y ) ) )
3230, 31syl 16 1  |-  ( ph  ->  ( X G Y ) : ( X H Y ) --> ( ( F `  X
) J ( F `
 Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4039   class class class wbr 4453    X. cxp 5003   -->wf 5590   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794    ^m cmap 7432   X_cixp 7481   Basecbs 14507   Hom chom 14583    Func cfunc 15098
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434  df-ixp 7482  df-func 15102
This theorem is referenced by:  funcsect  15116  funcoppc  15119  cofu2  15130  cofucl  15132  cofulid  15134  cofurid  15135  funcres  15140  funcres2  15142  funcres2c  15145  isfull2  15155  isfth2  15159  fthsect  15169  fthmon  15171  fuccocl  15208  fucidcl  15209  invfuc  15218  natpropd  15220  catciso  15309  prfval  15343  prfcl  15347  prf1st  15348  prf2nd  15349  1st2ndprf  15350  evlfcllem  15365  evlfcl  15366  curf1cl  15372  curf2cl  15375  uncf2  15381  curfuncf  15382  uncfcurf  15383  diag2cl  15390  curf2ndf  15391  yonedalem4c  15421  yonedalem3b  15423  yonedainv  15425  yonffthlem  15426
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