Theorem parhomvaln0 50844

Description: Evaluate an inhabited parallel-arrow hom-set. Membership certifies the selected branch even when sethood hypotheses are unavailable. Compare parhomval 50834 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 2922	. . . 4 ⊢ Ⅎ𝑎if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅))
3		nfcv 2922	. . . 4 ⊢ Ⅎ𝑏if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅))
4		nfcv 2922	. . . 4 ⊢ Ⅎ𝑥if(𝑎 = 𝑏, {𝐽}, if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅))
5		nfcv 2922	. . . 4 ⊢ Ⅎ𝑦if(𝑎 = 𝑏, {𝐽}, if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅))
6		eqeq12 2777	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 = 𝑦 ↔ 𝑎 = 𝑏))
7		simpl 488	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑎)
8	7	eqeq1d 2762	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 = 𝑋 ↔ 𝑎 = 𝑋))
9		simpr 490	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑦 = 𝑏)
10	9	eqeq1d 2762	. . . . . . 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 4505	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅) = if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅))
13	6, 12	ifbieq2d 4508	. . . 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 7514	. . 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 2783	. 2 ⊢ 𝐻 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ if(𝑎 = 𝑏, {𝐽}, if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅)))
16		eqeq12 2777	. . 3 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (𝑎 = 𝑏 ↔ 𝑈 = 𝑉))
17		simpl 488	. . . . . 6 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → 𝑎 = 𝑈)
18	17	eqeq1d 2762	. . . . 5 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (𝑎 = 𝑋 ↔ 𝑈 = 𝑋))
19		simpr 490	. . . . . 6 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → 𝑏 = 𝑉)
20	19	eqeq1d 2762	. . . . 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 4505	. . 3 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → if((𝑎 = 𝑋 ∧ 𝑏 = 𝑌), 𝑆, ∅) = if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅))
23	16, 22	ifbieq2d 4508	. 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 50071	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 1569   ∈ wcel 2144  ∅c0 4278  ifcif 4481  {csn 4583  (class class class)co 7420   ∈ cmpo 7422

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-dm 5660  df-iota 6494  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425

This theorem is used by:  parhomne  50846  parmapcl  50862