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Theorem funcfn2 13759
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 2296 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
3 eqid 2296 . . 3  |-  (  Hom  `  E )  =  (  Hom  `  E )
4 funcfn2.f . . 3  |-  ( ph  ->  F ( D  Func  E ) G )
51, 2, 3, 4funcixp 13757 . 2  |-  ( ph  ->  G  e.  X_ x  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  x ) ) (  Hom  `  E )
( F `  ( 2nd `  x ) ) )  ^m  ( (  Hom  `  D ) `  x ) ) )
6 ixpfn 6838 . 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 15 1  |-  ( ph  ->  G  Fn  ( B  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   class class class wbr 4039    X. cxp 4703    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137    ^m cmap 6788   X_cixp 6833   Basecbs 13164    Hom chom 13235    Func cfunc 13744
This theorem is referenced by:  funcoppc  13765  cofuval  13772  cofulid  13780  cofurid  13781  prf1st  13994  prf2nd  13995  1st2ndprf  13996  curfuncf  14028  uncfcurf  14029  curf2ndf  14037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-ixp 6834  df-func 13748
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