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Theorem homffn 14624
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 2438 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
41, 2, 3homffval 14622 . 2  |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x ( Hom  `  C ) y ) )
5 ovex 6111 . 2  |-  ( x ( Hom  `  C
) y )  e. 
_V
64, 5fnmpt2i 6638 1  |-  F  Fn  ( B  X.  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    X. cxp 4833    Fn wfn 5408   ` cfv 5413  (class class class)co 6086   Basecbs 14166   Hom chom 14241   Hom f chomf 14596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-homf 14600
This theorem is referenced by:  homfeqbas  14627  2oppchomf  14655  subcss1  14744  issubc3  14751  fullsubc  14752  fullresc  14753  funcres2c  14803  hofcllem  15060  hofcl  15061  oppchofcl  15062  oyoncl  15072  yonffthlem  15084
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