Theorem homffn 17899

Description: The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
homffn.f	⊢ 𝐹 = (Homf ‘𝐶)
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.

Step	Hyp	Ref	Expression
1		homffn.f	. . 3 ⊢ 𝐹 = (Homf ‘𝐶)
2		homffn.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2763	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	homffval 17896	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(Hom ‘𝐶)𝑦))
5		ovex 7479	. 2 ⊢ (𝑥(Hom ‘𝐶)𝑦) ∈ V
6	4, 5	fnmpoi 8109	1 ⊢ 𝐹 Fn (𝐵 × 𝐵)

This proof depends on syntax axioms:   = wceq 1570   × cxp 5654   Fn wfn 6556  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  Hom chom 17471  Hom_f chomf 17872

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-homf 17876

This theorem is used by:  homfeqbas  17902  2oppchomf  17930  0ssc  18044  catsubcat  18046  subcss1  18049  issubc3  18056  fullsubc  18057  fullresc  18058  funcres2c  18110  hofcllem  18464  hofcl  18465  oppchofcl  18466  oyoncl  18476  yonffthlem  18488  srhmsubc  20965  srhmsubcALTV  49637  homf0  50338  oppcendc  50347  discsubc  50393  nelsubclem  50396  ssccatid  50401  resccatlem  50402  imaidfu  50441  imaidfu2  50442  imasubc  50482  imassc  50484  setc1onsubc  50955