Theorem fpwwe2lem4 10629

Description: Lemma for fpwwe2 10638. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.)

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 482	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝐴 ∈ 𝑉)
3		simpr1 1195	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋 ⊆ 𝐴)
4	2, 3	ssexd 5325	. . 3 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋 ∈ V)
5	4, 4	xpexd 7738	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 × 𝑋) ∈ V)
6		simpr2 1196	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ⊆ (𝑋 × 𝑋))
7	5, 6	ssexd 5325	. . 3 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ∈ V)
8	4, 7	jca 513	. 2 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
9		sseq1 4008	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
10		xpeq12 5702	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
11	10	anidms 568	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 × 𝑥) = (𝑋 × 𝑋))
12	11	sseq2d 4015	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑋 × 𝑋)))
13		weeq2 5666	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑋))
14	9, 12, 13	3anbi123d 1437	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)))
15	14	anbi2d 630	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) ↔ (𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋))))
16		oveq1 7416	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥𝐹𝑟) = (𝑋𝐹𝑟))
17	16	eleq1d 2819	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑟) ∈ 𝐴))
18	15, 17	imbi12d 345	. . 3 ⊢ (𝑥 = 𝑋 → (((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)) → (𝑋𝐹𝑟) ∈ 𝐴)))
19		sseq1 4008	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 ⊆ (𝑋 × 𝑋) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
20		weeq1 5665	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 We 𝑋 ↔ 𝑅 We 𝑋))
21	19, 20	3anbi23d 1440	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)))
22	21	anbi2d 630	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)) ↔ (𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋))))
23		oveq2 7417	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑋𝐹𝑟) = (𝑋𝐹𝑅))
24	23	eleq1d 2819	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑋𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑅) ∈ 𝐴))
25	22, 24	imbi12d 345	. . 3 ⊢ (𝑟 = 𝑅 → (((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)) → (𝑋𝐹𝑟) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)))
26		fpwwe2.3	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
27	18, 25, 26	vtocl2g 3563	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑅 ∈ V) → ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴))
28	8, 27	mpcom 38	1 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475  [wsbc 3778   ∩ cin 3948   ⊆ wss 3949  {csn 4629  {copab 5211   We wwe 5631   × cxp 5675  ^◡ccnv 5676   “ cima 5680  (class class class)co 7409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-iota 6496  df-fv 6552  df-ov 7412

This theorem is referenced by:  fpwwe2lem12  10637