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Theorem homffn 14938
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 2460 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
41, 2, 3homffval 14936 . 2  |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x ( Hom  `  C ) y ) )
5 ovex 6300 . 2  |-  ( x ( Hom  `  C
) y )  e. 
_V
64, 5fnmpt2i 6843 1  |-  F  Fn  ( B  X.  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    X. cxp 4990    Fn wfn 5574   ` cfv 5579  (class class class)co 6275   Basecbs 14479   Hom chom 14555   Hom f chomf 14910
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-homf 14914
This theorem is referenced by:  homfeqbas  14941  2oppchomf  14969  subcss1  15058  issubc3  15065  fullsubc  15066  fullresc  15067  funcres2c  15117  hofcllem  15374  hofcl  15375  oppchofcl  15376  oyoncl  15386  yonffthlem  15398
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