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Theorem funcf2 15724
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 6308 . . . 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 15723 . . . . 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 4886 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
107, 8, 9syl2anc 665 . . . . 5  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
11 fveq2 5881 . . . . . . . . 9  |-  ( z  =  <. X ,  Y >.  ->  ( 1st `  z
)  =  ( 1st `  <. X ,  Y >. ) )
1211fveq2d 5885 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( F `  ( 1st `  z ) )  =  ( F `
 ( 1st `  <. X ,  Y >. )
) )
13 fveq2 5881 . . . . . . . . 9  |-  ( z  =  <. X ,  Y >.  ->  ( 2nd `  z
)  =  ( 2nd `  <. X ,  Y >. ) )
1413fveq2d 5885 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( F `  ( 2nd `  z ) )  =  ( F `
 ( 2nd `  <. X ,  Y >. )
) )
1512, 14oveq12d 6323 . . . . . . 7  |-  ( z  =  <. X ,  Y >.  ->  ( ( F `
 ( 1st `  z
) ) J ( F `  ( 2nd `  z ) ) )  =  ( ( F `
 ( 1st `  <. X ,  Y >. )
) J ( F `
 ( 2nd `  <. X ,  Y >. )
) ) )
16 fveq2 5881 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
17 df-ov 6308 . . . . . . . 8  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1816, 17syl6eqr 2488 . . . . . . 7  |-  ( z  =  <. X ,  Y >.  ->  ( H `  z )  =  ( X H Y ) )
1915, 18oveq12d 6323 . . . . . 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 7535 . . . . 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 665 . . . 4  |-  ( ph  ->  ( G `  <. X ,  Y >. )  e.  ( ( ( F `
 ( 1st `  <. X ,  Y >. )
) J ( F `
 ( 2nd `  <. X ,  Y >. )
) )  ^m  ( X H Y ) ) )
221, 21syl5eqel 2521 . . 3  |-  ( ph  ->  ( X G Y )  e.  ( ( ( F `  ( 1st `  <. X ,  Y >. ) ) J ( F `  ( 2nd `  <. X ,  Y >. ) ) )  ^m  ( X H Y ) ) )
23 op1stg 6819 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
2423fveq2d 5885 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( 1st `  <. X ,  Y >. ) )  =  ( F `  X ) )
25 op2ndg 6820 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2625fveq2d 5885 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( 2nd `  <. X ,  Y >. ) )  =  ( F `  Y ) )
2724, 26oveq12d 6323 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( F `  ( 1st `  <. X ,  Y >. ) ) J ( F `  ( 2nd `  <. X ,  Y >. ) ) )  =  ( ( F `  X ) J ( F `  Y ) ) )
287, 8, 27syl2anc 665 . . . 4  |-  ( ph  ->  ( ( F `  ( 1st `  <. X ,  Y >. ) ) J ( F `  ( 2nd `  <. X ,  Y >. ) ) )  =  ( ( F `  X ) J ( F `  Y ) ) )
2928oveq1d 6320 . . 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 2519 . 2  |-  ( ph  ->  ( X G Y )  e.  ( ( ( F `  X
) J ( F `
 Y ) )  ^m  ( X H Y ) ) )
31 elmapi 7501 . 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 17 1  |-  ( ph  ->  ( X G Y ) : ( X H Y ) --> ( ( F `  X
) J ( F `
 Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   <.cop 4008   class class class wbr 4426    X. cxp 4852   -->wf 5597   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806    ^m cmap 7480   X_cixp 7530   Basecbs 15084   Hom chom 15163    Func cfunc 15710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-ixp 7531  df-func 15714
This theorem is referenced by:  funcsect  15728  funcoppc  15731  cofu2  15742  cofucl  15744  cofulid  15746  cofurid  15747  funcres  15752  funcres2  15754  funcres2c  15757  isfull2  15767  isfth2  15771  fthsect  15781  fthmon  15783  fuccocl  15820  fucidcl  15821  invfuc  15830  natpropd  15832  catciso  15953  prfval  16035  prfcl  16039  prf1st  16040  prf2nd  16041  1st2ndprf  16042  evlfcllem  16057  evlfcl  16058  curf1cl  16064  curf2cl  16067  uncf2  16073  curfuncf  16074  uncfcurf  16075  diag2cl  16082  curf2ndf  16083  yonedalem4c  16113  yonedalem3b  16115  yonedainv  16117  yonffthlem  16118
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