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Theorem homffn 16554
 Description: The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
homffn.f 𝐹 = (Homf𝐶)
homffn.b 𝐵 = (Base‘𝐶)
Assertion
Ref Expression
homffn 𝐹 Fn (𝐵 × 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 = (Homf𝐶)
2 homffn.b . . 3 𝐵 = (Base‘𝐶)
3 eqid 2760 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
41, 2, 3homffval 16551 . 2 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥(Hom ‘𝐶)𝑦))
5 ovex 6841 . 2 (𝑥(Hom ‘𝐶)𝑦) ∈ V
64, 5fnmpt2i 7407 1 𝐹 Fn (𝐵 × 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   × cxp 5264   Fn wfn 6044  ‘cfv 6049  (class class class)co 6813  Basecbs 16059  Hom chom 16154  Homf chomf 16528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-homf 16532 This theorem is referenced by:  homfeqbas  16557  2oppchomf  16585  0ssc  16698  catsubcat  16700  subcss1  16703  issubc3  16710  fullsubc  16711  fullresc  16712  funcres2c  16762  hofcllem  17099  hofcl  17100  oppchofcl  17101  oyoncl  17111  yonffthlem  17123  srhmsubc  42586  srhmsubcALTV  42604
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