Theorem fpwwe2lem5 9335

Description: Lemma for fpwwe2 9344. (Contributed by Mario Carneiro, 15-May-2015.)

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
fpwwe2lem5	⊢ ((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)

Proof of Theorem fpwwe2lem5

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 1060	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋 ⊆ 𝐴)
4	2, 3	ssexd 4733	. . 3 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋 ∈ V)
5		xpexg 6858	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
6	4, 4, 5	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 × 𝑋) ∈ V)
7		simpr2 1061	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ⊆ (𝑋 × 𝑋))
8	6, 7	ssexd 4733	. . 3 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ∈ V)
9	4, 8	jca 553	. 2 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
10		sseq1 3589	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
11		xpeq12 5058	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
12	11	anidms 675	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 × 𝑥) = (𝑋 × 𝑋))
13	12	sseq2d 3596	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑋 × 𝑋)))
14		weeq2 5027	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑋))
15	10, 13, 14	3anbi123d 1391	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)))
16	15	anbi2d 736	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) ↔ (𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋))))
17		oveq1 6556	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥𝐹𝑟) = (𝑋𝐹𝑟))
18	17	eleq1d 2672	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑟) ∈ 𝐴))
19	16, 18	imbi12d 333	. . 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)))
20		sseq1 3589	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 ⊆ (𝑋 × 𝑋) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
21		weeq1 5026	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 We 𝑋 ↔ 𝑅 We 𝑋))
22	20, 21	3anbi23d 1394	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)))
23	22	anbi2d 736	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)) ↔ (𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋))))
24		oveq2 6557	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑋𝐹𝑟) = (𝑋𝐹𝑅))
25	24	eleq1d 2672	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑋𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑅) ∈ 𝐴))
26	23, 25	imbi12d 333	. . 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)))
27		fpwwe2.3	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
28	19, 26, 27	vtocl2g 3243	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑅 ∈ V) → ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴))
29	9, 28	mpcom 37	1 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402   ∩ cin 3539   ⊆ wss 3540  {csn 4125  {copab 4642   We wwe 4996   × cxp 5036  ^◡ccnv 5037   “ cima 5041  (class class class)co 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-iota 5768  df-fv 5812  df-ov 6552

This theorem is referenced by:  fpwwe2lem13  9343