Theorem fpwwe2lem4 10557

Description: Lemma for fpwwe2 10566. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.) (Proof shortened by Matthew House, 10-Sep-2025.)

Hypotheses

Ref	Expression
fpwwe2.1	⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fpwwe2.3	⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)

Assertion

Ref	Expression
fpwwe2lem4	⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)

Distinct variable groups:   𝑦,𝑢,𝑟,𝑥,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑅,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑦,𝑢)   𝑉(𝑥,𝑦,𝑢,𝑟)

Proof of Theorem fpwwe2lem4

Step	Hyp	Ref	Expression
1		fpwwe2.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝐴 ∈ 𝑉)
3		simpr1 1196	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋 ⊆ 𝐴)
4	2, 3	ssexd 5271	. . 3 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋 ∈ V)
5	4, 4	xpexd 7706	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 × 𝑋) ∈ V)
6		simpr2 1197	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ⊆ (𝑋 × 𝑋))
7	5, 6	ssexd 5271	. . 3 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ∈ V)
8		simpl 482	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → 𝑥 = 𝑋)
9	8	sseq1d 3967	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
10		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
11	8	sqxpeqd 5664	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
12	10, 11	sseq12d 3969	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
13	10, 8	weeq12d 5621	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑟 We 𝑥 ↔ 𝑅 We 𝑋))
14	9, 12, 13	3anbi123d 1439	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)))
15		oveq12 7377	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑥𝐹𝑟) = (𝑋𝐹𝑅))
16	15	eleq1d 2822	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑥𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑅) ∈ 𝐴))
17	14, 16	imbi12d 344	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥𝐹𝑟) ∈ 𝐴) ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋) → (𝑋𝐹𝑅) ∈ 𝐴)))
18		fpwwe2.3	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
19	18	ex 412	. . . 4 ⊢ (𝜑 → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥𝐹𝑟) ∈ 𝐴))
20	19	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥𝐹𝑟) ∈ 𝐴))
21	4, 7, 17, 20	vtocl2d 3521	. 2 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋) → (𝑋𝐹𝑅) ∈ 𝐴))
22	21	syldbl2 842	1 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442  [wsbc 3742   ∩ cin 3902   ⊆ wss 3903  {csn 4582  {copab 5162   We wwe 5584   × cxp 5630  ^◡ccnv 5631   “ cima 5635  (class class class)co 7368

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-iota 6456  df-fv 6508  df-ov 7371

This theorem is referenced by:  fpwwe2lem12  10565