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Theorem funcf2 14770
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 6089 . . . 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 14769 . . . . 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 4866 . . . . . 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 5686 . . . . . . . . 9  |-  ( z  =  <. X ,  Y >.  ->  ( 1st `  z
)  =  ( 1st `  <. X ,  Y >. ) )
1211fveq2d 5690 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( F `  ( 1st `  z ) )  =  ( F `
 ( 1st `  <. X ,  Y >. )
) )
13 fveq2 5686 . . . . . . . . 9  |-  ( z  =  <. X ,  Y >.  ->  ( 2nd `  z
)  =  ( 2nd `  <. X ,  Y >. ) )
1413fveq2d 5690 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( F `  ( 2nd `  z ) )  =  ( F `
 ( 2nd `  <. X ,  Y >. )
) )
1512, 14oveq12d 6104 . . . . . . 7  |-  ( z  =  <. X ,  Y >.  ->  ( ( F `
 ( 1st `  z
) ) J ( F `  ( 2nd `  z ) ) )  =  ( ( F `
 ( 1st `  <. X ,  Y >. )
) J ( F `
 ( 2nd `  <. X ,  Y >. )
) ) )
16 fveq2 5686 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
17 df-ov 6089 . . . . . . . 8  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1816, 17syl6eqr 2488 . . . . . . 7  |-  ( z  =  <. X ,  Y >.  ->  ( H `  z )  =  ( X H Y ) )
1915, 18oveq12d 6104 . . . . . 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 7260 . . . . 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 2522 . . 3  |-  ( ph  ->  ( X G Y )  e.  ( ( ( F `  ( 1st `  <. X ,  Y >. ) ) J ( F `  ( 2nd `  <. X ,  Y >. ) ) )  ^m  ( X H Y ) ) )
23 op1stg 6584 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
2423fveq2d 5690 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( 1st `  <. X ,  Y >. ) )  =  ( F `  X ) )
25 op2ndg 6585 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2625fveq2d 5690 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( 2nd `  <. X ,  Y >. ) )  =  ( F `  Y ) )
2724, 26oveq12d 6104 . . . . 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 6101 . . 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 2514 . 2  |-  ( ph  ->  ( X G Y )  e.  ( ( ( F `  X
) J ( F `
 Y ) )  ^m  ( X H Y ) ) )
31 elmapi 7226 . 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 1369    e. wcel 1756   <.cop 3878   class class class wbr 4287    X. cxp 4833   -->wf 5409   ` cfv 5413  (class class class)co 6086   1stc1st 6570   2ndc2nd 6571    ^m cmap 7206   X_cixp 7255   Basecbs 14166   Hom chom 14241    Func cfunc 14756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-map 7208  df-ixp 7256  df-func 14760
This theorem is referenced by:  funcsect  14774  funcoppc  14777  cofu2  14788  cofucl  14790  cofulid  14792  cofurid  14793  funcres  14798  funcres2  14800  funcres2c  14803  isfull2  14813  isfth2  14817  fthsect  14827  fthmon  14829  fuccocl  14866  fucidcl  14867  invfuc  14876  natpropd  14878  catciso  14967  prfval  15001  prfcl  15005  prf1st  15006  prf2nd  15007  1st2ndprf  15008  evlfcllem  15023  evlfcl  15024  curf1cl  15030  curf2cl  15033  uncf2  15039  curfuncf  15040  uncfcurf  15041  diag2cl  15048  curf2ndf  15049  yonedalem4c  15079  yonedalem3b  15081  yonedainv  15083  yonffthlem  15084
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