Theorem homffn 17705

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 2735	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	homffval 17702	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(Hom ‘𝐶)𝑦))
5		ovex 7438	. 2 ⊢ (𝑥(Hom ‘𝐶)𝑦) ∈ V
6	4, 5	fnmpoi 8069	1 ⊢ 𝐹 Fn (𝐵 × 𝐵)

Syntax hints:   = wceq 1540   × cxp 5652   Fn wfn 6526  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  Hom chom 17282  Hom_f chomf 17678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-homf 17682

This theorem is referenced by:  homfeqbas  17708  2oppchomf  17736  0ssc  17850  catsubcat  17852  subcss1  17855  issubc3  17862  fullsubc  17863  fullresc  17864  funcres2c  17916  hofcllem  18270  hofcl  18271  oppchofcl  18272  oyoncl  18282  yonffthlem  18294  srhmsubc  20640  srhmsubcALTV  48300  homf0  48984  oppcendc  48993  discsubc  49031  nelsubclem  49034  ssccatid  49039  resccatlem  49040  imaidfu  49069  imaidfu2  49070  imasubc  49091  imassc  49093  setc1onsubc  49479