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Theorem funcf2 14900
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 6206 . . . 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 14899 . . . . 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 4982 . . . . . 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 5802 . . . . . . . . 9  |-  ( z  =  <. X ,  Y >.  ->  ( 1st `  z
)  =  ( 1st `  <. X ,  Y >. ) )
1211fveq2d 5806 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( F `  ( 1st `  z ) )  =  ( F `
 ( 1st `  <. X ,  Y >. )
) )
13 fveq2 5802 . . . . . . . . 9  |-  ( z  =  <. X ,  Y >.  ->  ( 2nd `  z
)  =  ( 2nd `  <. X ,  Y >. ) )
1413fveq2d 5806 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( F `  ( 2nd `  z ) )  =  ( F `
 ( 2nd `  <. X ,  Y >. )
) )
1512, 14oveq12d 6221 . . . . . . 7  |-  ( z  =  <. X ,  Y >.  ->  ( ( F `
 ( 1st `  z
) ) J ( F `  ( 2nd `  z ) ) )  =  ( ( F `
 ( 1st `  <. X ,  Y >. )
) J ( F `
 ( 2nd `  <. X ,  Y >. )
) ) )
16 fveq2 5802 . . . . . . . 8  |-  ( z  =  <. X ,  Y >.  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
17 df-ov 6206 . . . . . . . 8  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1816, 17syl6eqr 2513 . . . . . . 7  |-  ( z  =  <. X ,  Y >.  ->  ( H `  z )  =  ( X H Y ) )
1915, 18oveq12d 6221 . . . . . 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 7381 . . . . 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 2546 . . 3  |-  ( ph  ->  ( X G Y )  e.  ( ( ( F `  ( 1st `  <. X ,  Y >. ) ) J ( F `  ( 2nd `  <. X ,  Y >. ) ) )  ^m  ( X H Y ) ) )
23 op1stg 6702 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
2423fveq2d 5806 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( 1st `  <. X ,  Y >. ) )  =  ( F `  X ) )
25 op2ndg 6703 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2625fveq2d 5806 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( 2nd `  <. X ,  Y >. ) )  =  ( F `  Y ) )
2724, 26oveq12d 6221 . . . . 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 6218 . . 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 2544 . 2  |-  ( ph  ->  ( X G Y )  e.  ( ( ( F `  X
) J ( F `
 Y ) )  ^m  ( X H Y ) ) )
31 elmapi 7347 . 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 1370    e. wcel 1758   <.cop 3994   class class class wbr 4403    X. cxp 4949   -->wf 5525   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689    ^m cmap 7327   X_cixp 7376   Basecbs 14295   Hom chom 14371    Func cfunc 14886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-map 7329  df-ixp 7377  df-func 14890
This theorem is referenced by:  funcsect  14904  funcoppc  14907  cofu2  14918  cofucl  14920  cofulid  14922  cofurid  14923  funcres  14928  funcres2  14930  funcres2c  14933  isfull2  14943  isfth2  14947  fthsect  14957  fthmon  14959  fuccocl  14996  fucidcl  14997  invfuc  15006  natpropd  15008  catciso  15097  prfval  15131  prfcl  15135  prf1st  15136  prf2nd  15137  1st2ndprf  15138  evlfcllem  15153  evlfcl  15154  curf1cl  15160  curf2cl  15163  uncf2  15169  curfuncf  15170  uncfcurf  15171  diag2cl  15178  curf2ndf  15179  yonedalem4c  15209  yonedalem3b  15211  yonedainv  15213  yonffthlem  15214
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