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Theorem homffn 15099
Description: The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
homffn.f  |-  F  =  ( Hom f  `  C )
homffn.b  |-  B  =  ( Base `  C
)
Assertion
Ref Expression
homffn  |-  F  Fn  ( B  X.  B
)

Proof of Theorem homffn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 homffn.f . . 3  |-  F  =  ( Hom f  `  C )
2 homffn.b . . 3  |-  B  =  ( Base `  C
)
3 eqid 2382 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
41, 2, 3homffval 15096 . 2  |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x ( Hom  `  C ) y ) )
5 ovex 6224 . 2  |-  ( x ( Hom  `  C
) y )  e. 
_V
64, 5fnmpt2i 6768 1  |-  F  Fn  ( B  X.  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399    X. cxp 4911    Fn wfn 5491   ` cfv 5496  (class class class)co 6196   Basecbs 14634   Hom chom 14713   Hom f chomf 15073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-homf 15077
This theorem is referenced by:  homfeqbas  15102  2oppchomf  15130  0ssc  15243  catsubcat  15245  subcss1  15248  issubc3  15255  fullsubc  15256  fullresc  15257  funcres2c  15307  hofcllem  15644  hofcl  15645  oppchofcl  15646  oyoncl  15656  yonffthlem  15668  srhmsubc  33084  srhmsubcALTV  33103
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