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Theorem funcfn2 14775
Description: The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
funcfn2.b  |-  B  =  ( Base `  D
)
funcfn2.f  |-  ( ph  ->  F ( D  Func  E ) G )
Assertion
Ref Expression
funcfn2  |-  ( ph  ->  G  Fn  ( B  X.  B ) )

Proof of Theorem funcfn2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funcfn2.b . . 3  |-  B  =  ( Base `  D
)
2 eqid 2441 . . 3  |-  ( Hom  `  D )  =  ( Hom  `  D )
3 eqid 2441 . . 3  |-  ( Hom  `  E )  =  ( Hom  `  E )
4 funcfn2.f . . 3  |-  ( ph  ->  F ( D  Func  E ) G )
51, 2, 3, 4funcixp 14773 . 2  |-  ( ph  ->  G  e.  X_ x  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  x ) ) ( Hom  `  E )
( F `  ( 2nd `  x ) ) )  ^m  ( ( Hom  `  D ) `  x ) ) )
6 ixpfn 7265 . 2  |-  ( G  e.  X_ x  e.  ( B  X.  B ) ( ( ( F `
 ( 1st `  x
) ) ( Hom  `  E ) ( F `
 ( 2nd `  x
) ) )  ^m  ( ( Hom  `  D
) `  x )
)  ->  G  Fn  ( B  X.  B
) )
75, 6syl 16 1  |-  ( ph  ->  G  Fn  ( B  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761   class class class wbr 4289    X. cxp 4834    Fn wfn 5410   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575    ^m cmap 7210   X_cixp 7259   Basecbs 14170   Hom chom 14245    Func cfunc 14760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7212  df-ixp 7260  df-func 14764
This theorem is referenced by:  funcoppc  14781  cofuval  14788  cofulid  14796  cofurid  14797  prf1st  15010  prf2nd  15011  1st2ndprf  15012  curfuncf  15044  uncfcurf  15045  curf2ndf  15053
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