Theorem parhomco2 50965

Description: Variant of parhomco 50966. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
parhomval.h	⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
parhomeqor.f	⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
parhomeqor.g	⊢ (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))
parhomco.p	⊢ 𝑆 ⊆ 𝒫 𝐽
parhomco2.s	⊢ 𝑆 ∈ V
parhomco2.j	⊢ 𝐽 ∈ V
parhomco2.n	⊢ 𝑋 ≠ 𝑌
parhomco2.u	⊢ (𝜑 → 𝑈 ∈ 𝐵)
parhomco2.v	⊢ (𝜑 → 𝑉 ∈ 𝐵)
parhomco2.w	⊢ (𝜑 → 𝑊 ∈ 𝐵)

Assertion

Ref	Expression
parhomco2	⊢ (𝜑 → (𝐺 ∩ 𝐹) ∈ (𝑈𝐻𝑊))

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

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

Proof of Theorem parhomco2

Step	Hyp	Ref	Expression
1		parhomeqor.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))
2	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐺 ∈ (𝑉𝐻𝑊))
3		parhomval.h	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
4		parhomeqor.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
5	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐹 ∈ (𝑈𝐻𝑉))
6		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝑈 = 𝑉)
7	3, 5, 6	parhomideq 50955	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐹 = 𝐽)
8	7	ineq2d 4120	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺 ∩ 𝐹) = (𝐺 ∩ 𝐽))
9		parhomco.p	. . . . . . 7 ⊢ 𝑆 ⊆ 𝒫 𝐽
10	3, 9, 1	parhomss 50962	. . . . . 6 ⊢ (𝜑 → 𝐺 ⊆ 𝐽)
11	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐺 ⊆ 𝐽)
12		dfss2 4097	. . . . 5 ⊢ (𝐺 ⊆ 𝐽 ↔ (𝐺 ∩ 𝐽) = 𝐺)
13	11, 12	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺 ∩ 𝐽) = 𝐺)
14	8, 13	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺 ∩ 𝐹) = 𝐺)
15	6	oveq1d 7458	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝑈𝐻𝑊) = (𝑉𝐻𝑊))
16	2, 14, 15	3eltr4d 2879	. 2 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺 ∩ 𝐹) ∈ (𝑈𝐻𝑊))
17	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐺 ∈ (𝑉𝐻𝑊))
18		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝑉 = 𝑊)
19	3, 17, 18	parhomideq 50955	. . . . 5 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐺 = 𝐽)
20	19	ineq1d 4119	. . . 4 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐺 ∩ 𝐹) = (𝐽 ∩ 𝐹))
21	3, 9, 4	parhomss 50962	. . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ 𝐽)
22	21	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐹 ⊆ 𝐽)
23		sseqin2 4123	. . . . 5 ⊢ (𝐹 ⊆ 𝐽 ↔ (𝐽 ∩ 𝐹) = 𝐹)
24	22, 23	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐽 ∩ 𝐹) = 𝐹)
25	20, 24	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐺 ∩ 𝐹) = 𝐹)
26	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐹 ∈ (𝑈𝐻𝑉))
27	18	oveq2d 7459	. . . 4 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝑈𝐻𝑉) = (𝑈𝐻𝑊))
28	26, 27	eleqtrd 2866	. . 3 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐹 ∈ (𝑈𝐻𝑊))
29	25, 28	eqeltrd 2864	. 2 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐺 ∩ 𝐹) ∈ (𝑈𝐻𝑊))
30	3, 4, 1	parhomeqor 50964	. 2 ⊢ (𝜑 → (𝑈 = 𝑉 ∨ 𝑉 = 𝑊))
31	16, 29, 30	mpjaodan 973	1 ⊢ (𝜑 → (𝐺 ∩ 𝐹) ∈ (𝑈𝐻𝑊))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  ifcif 4485  𝒫 cpw 4560  {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-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448

This theorem is used by: (None)