Theorem homffn 17847

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 2761	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4	1, 2, 3	homffval 17844	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(Hom ‘𝐶)𝑦))
5		ovex 7445	. 2 ⊢ (𝑥(Hom ‘𝐶)𝑦) ∈ V
6	4, 5	fnmpoi 8070	1 ⊢ 𝐹 Fn (𝐵 × 𝐵)

This proof depends on syntax axioms:   = wceq 1570   × cxp 5649   Fn wfn 6526  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  Hom chom 17419  Hom_f chomf 17820

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-homf 17824

This theorem is used by:  homfeqbas  17850  2oppchomf  17878  0ssc  17992  catsubcat  17994  subcss1  17997  issubc3  18004  fullsubc  18005  fullresc  18006  funcres2c  18058  hofcllem  18412  hofcl  18413  oppchofcl  18414  oyoncl  18424  yonffthlem  18436  srhmsubc  20912  srhmsubcALTV  49366  homf0  50061  oppcendc  50070  discsubc  50116  nelsubclem  50119  ssccatid  50124  resccatlem  50125  imaidfu  50162  imaidfu2  50163  imasubc  50203  imassc  50205  setc1onsubc  50654