Theorem homffn 17319

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 2738	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	homffval 17316	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(Hom ‘𝐶)𝑦))
5		ovex 7288	. 2 ⊢ (𝑥(Hom ‘𝐶)𝑦) ∈ V
6	4, 5	fnmpoi 7883	1 ⊢ 𝐹 Fn (𝐵 × 𝐵)

Syntax hints:   = wceq 1539   × cxp 5578   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  Hom_f chomf 17292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-homf 17296

This theorem is referenced by:  homfeqbas  17322  2oppchomf  17352  0ssc  17468  catsubcat  17470  subcss1  17473  issubc3  17480  fullsubc  17481  fullresc  17482  funcres2c  17533  hofcllem  17892  hofcl  17893  oppchofcl  17894  oyoncl  17904  yonffthlem  17916  srhmsubc  45522  srhmsubcALTV  45540