Theorem homffn 17751

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 2740	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	homffval 17748	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(Hom ‘𝐶)𝑦))
5		ovex 7481	. 2 ⊢ (𝑥(Hom ‘𝐶)𝑦) ∈ V
6	4, 5	fnmpoi 8111	1 ⊢ 𝐹 Fn (𝐵 × 𝐵)

Syntax hints:   = wceq 1537   × cxp 5698   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  Hom chom 17322  Hom_f chomf 17724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-homf 17728

This theorem is referenced by:  homfeqbas  17754  2oppchomf  17784  0ssc  17901  catsubcat  17903  subcss1  17906  issubc3  17913  fullsubc  17914  fullresc  17915  funcres2c  17968  hofcllem  18328  hofcl  18329  oppchofcl  18330  oyoncl  18340  yonffthlem  18352  srhmsubc  20702  srhmsubcALTV  48048