Theorem homffval 17747

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 6883	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		homffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eqtr4di 2816	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5		fveq2 6883	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6		homffval.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶)
7	5, 6	eqtr4di 2816	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
8	7	oveqd 7429	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦))
9	4, 4, 8	mpoeq123dv 7487	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
10		df-homf 17727	. . . 4 ⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
11	3	fvexi 6897	. . . . 5 ⊢ 𝐵 ∈ V
12	11, 11	mpoex 8077	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) ∈ V
13	9, 10, 12	fvmpt 6991	. . 3 ⊢ (𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
14		fvprc 6875	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = ∅)
15		fvprc 6875	. . . . . . 7 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
16	3, 15	eqtrid 2810	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → 𝐵 = ∅)
17	16	olcd 887	. . . . 5 ⊢ (¬ 𝐶 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
18		0mpo0 7495	. . . . 5 ⊢ ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = ∅)
19	17, 18	syl 18	. . . 4 ⊢ (¬ 𝐶 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = ∅)
20	14, 19	eqtr4d 2801	. . 3 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
21	13, 20	pm2.61i 184	. 2 ⊢ (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
22	1, 21	eqtri 2786	1 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))

Syntax hints:  ¬ wn 3   ∨ wo 860   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4287  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414  Basecbs 17270  Hom chom 17322  Hom_f chomf 17723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-homf 17727

This theorem is referenced by:  fnhomeqhomf  17748  homfval  17749  homffn  17750  homfeq  17751  oppchomf  17777  reschomf  17889  homf0  49764