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Theorem homffn 14615
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 2433 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
41, 2, 3homffval 14613 . 2  |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x ( Hom  `  C ) y ) )
5 ovex 6105 . 2  |-  ( x ( Hom  `  C
) y )  e. 
_V
64, 5fnmpt2i 6632 1  |-  F  Fn  ( B  X.  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1362    X. cxp 4825    Fn wfn 5401   ` cfv 5406  (class class class)co 6080   Basecbs 14157   Hom chom 14232   Hom f chomf 14587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-homf 14591
This theorem is referenced by:  homfeqbas  14618  2oppchomf  14646  subcss1  14735  issubc3  14742  fullsubc  14743  fullresc  14744  funcres2c  14794  hofcllem  15051  hofcl  15052  oppchofcl  15053  oyoncl  15063  yonffthlem  15075
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