Theorem parhomvaln0 50958

Description: Evaluate an inhabited parallel-arrow hom-set. Membership certifies the selected branch even when sethood hypotheses are unavailable. Compare parhomval 50948 for evaluation throughout the object domain. (Contributed by Zhi Wang, 17-Sep-2026.)

Hypothesis

Ref	Expression
parhomval.h	⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))

Assertion

Ref	Expression
parhomvaln0	⊢ (𝐹 ∈ (𝑈𝐻𝑉) → (𝑈𝐻𝑉) = if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐽,𝑦   𝑥,𝑆,𝑦   𝑥,𝑈,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑉,𝑦

Allowed substitution hints:   𝐹(𝑥, 𝑦)   𝐻(𝑥, 𝑦)

Proof of Theorem parhomvaln0

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		parhomval.h	. . 3 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
2		nfcv 2926	. . . 4 ⊢ Ⅎ𝑎if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅))
3		nfcv 2926	. . . 4 ⊢ Ⅎ𝑏if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅))
4		nfcv 2926	. . . 4 ⊢ Ⅎ𝑥if(𝑎 = 𝑏, {𝐽}, if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅))
5		nfcv 2926	. . . 4 ⊢ Ⅎ𝑦if(𝑎 = 𝑏, {𝐽}, if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅))
6		eqeq12 2780	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 = 𝑦 ↔ 𝑎 = 𝑏))
7		simpl 488	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑎)
8	7	eqeq1d 2765	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 = 𝑋 ↔ 𝑎 = 𝑋))
9		simpr 490	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑦 = 𝑏)
10	9	eqeq1d 2765	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑦 = 𝑌 ↔ 𝑏 = 𝑌))
11	8, 10	anbi12d 644	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) ↔ (𝑎 = 𝑋 ∧ 𝑏 = 𝑌)))
12	11	ifbid 4509	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅) = if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅))
13	6, 12	ifbieq2d 4512	. . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)) = if(𝑎 = 𝑏, {𝐽}, if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅)))
14	2, 3, 4, 5, 13	cbvmpo 7537	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅))) = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ if(𝑎 = 𝑏, {𝐽}, if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅)))
15	1, 14	eqtri 2786	. 2 ⊢ 𝐻 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ if(𝑎 = 𝑏, {𝐽}, if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅)))
16		eqeq12 2780	. . 3 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (𝑎 = 𝑏 ↔ 𝑈 = 𝑉))
17		simpl 488	. . . . . 6 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → 𝑎 = 𝑈)
18	17	eqeq1d 2765	. . . . 5 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (𝑎 = 𝑋 ↔ 𝑈 = 𝑋))
19		simpr 490	. . . . . 6 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → 𝑏 = 𝑉)
20	19	eqeq1d 2765	. . . . 5 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (𝑏 = 𝑌 ↔ 𝑉 = 𝑌))
21	18, 20	anbi12d 644	. . . 4 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) ↔ (𝑈 = 𝑋 ∧ 𝑉 = 𝑌)))
22	21	ifbid 4509	. . 3 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅) = if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅))
23	16, 22	ifbieq2d 4512	. 2 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → if(𝑎 = 𝑏, {𝐽}, if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅)) = if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)))
24	15, 23	elovmpoeq 50185	1 ⊢ (𝐹 ∈ (𝑈𝐻𝑉) → (𝑈𝐻𝑉) = if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∅c0 4281  ifcif 4485  {csn 4588  (class class class)co 7443   ∈ cmpo 7445

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-dm 5665  df-iota 6512  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448

This theorem is used by:  parhomne  50960  parmapcl  50976