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Theorem funcf2 14070
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 6087 . . . 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 14069 . . . . 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 4913 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
107, 8, 9syl2anc 644 . . . . 5  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
11 fveq2 5731 . . . . . . . . 9  |-  ( z  =  <. X ,  Y >.  ->  ( 1st `  z
)  =  ( 1st `  <. X ,  Y >. ) )
1211fveq2d 5735 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( F `  ( 1st `  z ) )  =  ( F `
 ( 1st `  <. X ,  Y >. )
) )
13 fveq2 5731 . . . . . . . . 9  |-  ( z  =  <. X ,  Y >.  ->  ( 2nd `  z
)  =  ( 2nd `  <. X ,  Y >. ) )
1413fveq2d 5735 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( F `  ( 2nd `  z ) )  =  ( F `
 ( 2nd `  <. X ,  Y >. )
) )
1512, 14oveq12d 6102 . . . . . . 7  |-  ( z  =  <. X ,  Y >.  ->  ( ( F `
 ( 1st `  z
) ) J ( F `  ( 2nd `  z ) ) )  =  ( ( F `
 ( 1st `  <. X ,  Y >. )
) J ( F `
 ( 2nd `  <. X ,  Y >. )
) ) )
16 fveq2 5731 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
17 df-ov 6087 . . . . . . . 8  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1816, 17syl6eqr 2488 . . . . . . 7  |-  ( z  =  <. X ,  Y >.  ->  ( H `  z )  =  ( X H Y ) )
1915, 18oveq12d 6102 . . . . . 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 7070 . . . . 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 644 . . . 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 6362 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
2423fveq2d 5735 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( 1st `  <. X ,  Y >. ) )  =  ( F `  X ) )
25 op2ndg 6363 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2625fveq2d 5735 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( 2nd `  <. X ,  Y >. ) )  =  ( F `  Y ) )
2724, 26oveq12d 6102 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( F `  ( 1st `  <. X ,  Y >. ) ) J ( F `  ( 2nd `  <. X ,  Y >. ) ) )  =  ( ( F `  X ) J ( F `  Y ) ) )
287, 8, 27syl2anc 644 . . . 4  |-  ( ph  ->  ( ( F `  ( 1st `  <. X ,  Y >. ) ) J ( F `  ( 2nd `  <. X ,  Y >. ) ) )  =  ( ( F `  X ) J ( F `  Y ) ) )
2928oveq1d 6099 . . 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 7041 . 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   <.cop 3819   class class class wbr 4215    X. cxp 4879   -->wf 5453   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351    ^m cmap 7021   X_cixp 7066   Basecbs 13474    Hom chom 13545    Func cfunc 14056
This theorem is referenced by:  funcsect  14074  funcoppc  14077  cofu2  14088  cofucl  14090  cofulid  14092  cofurid  14093  funcres  14098  funcres2  14100  funcres2c  14103  isfull2  14113  isfth2  14117  fthsect  14127  fthmon  14129  fuccocl  14166  fucidcl  14167  invfuc  14176  natpropd  14178  catciso  14267  prfval  14301  prfcl  14305  prf1st  14306  prf2nd  14307  1st2ndprf  14308  evlfcllem  14323  evlfcl  14324  curf1cl  14330  curf2cl  14333  uncf2  14339  curfuncf  14340  uncfcurf  14341  diag2cl  14348  curf2ndf  14349  yonedalem4c  14379  yonedalem3b  14381  yonedainv  14383  yonffthlem  14384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-map 7023  df-ixp 7067  df-func 14060
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