Theorem homffval 16952

Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)

Hypotheses

Ref	Expression
homffval.f	⊢ 𝐹 = (Homf ‘𝐶)
homffval.b	⊢ 𝐵 = (Base‘𝐶)
homffval.h	⊢ 𝐻 = (Hom ‘𝐶)

Assertion

Ref	Expression
homffval	⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐻,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem homffval

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		homffval.f	. 2 ⊢ 𝐹 = (Homf ‘𝐶)
2		fveq2 6645	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		homffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eqtr4di 2851	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5		fveq2 6645	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6		homffval.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶)
7	5, 6	eqtr4di 2851	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
8	7	oveqd 7152	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦))
9	4, 4, 8	mpoeq123dv 7208	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
10		df-homf 16933	. . . 4 ⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
11	3	fvexi 6659	. . . . 5 ⊢ 𝐵 ∈ V
12	11, 11	mpoex 7760	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) ∈ V
13	9, 10, 12	fvmpt 6745	. . 3 ⊢ (𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
14		fvprc 6638	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = ∅)
15		fvprc 6638	. . . . . . 7 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
16	3, 15	syl5eq 2845	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → 𝐵 = ∅)
17	16	olcd 871	. . . . 5 ⊢ (¬ 𝐶 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
18		0mpo0 7216	. . . . 5 ⊢ ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = ∅)
19	17, 18	syl 17	. . . 4 ⊢ (¬ 𝐶 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = ∅)
20	14, 19	eqtr4d 2836	. . 3 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
21	13, 20	pm2.61i 185	. 2 ⊢ (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
22	1, 21	eqtri 2821	1 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))

Syntax hints:  ¬ wn 3   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  Basecbs 16475  Hom chom 16568  Hom_f chomf 16929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-homf 16933

This theorem is referenced by:  fnhomeqhomf  16953  homfval  16954  homffn  16955  homfeq  16956  oppchomf  16982  reschomf  17093