Theorem fpwwe2lem5 10058

Description: Lemma for fpwwe2 10067. (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 483	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝐴 ∈ V)
3		simpr1 1190	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋 ⊆ 𝐴)
4	2, 3	ssexd 5230	. . 3 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋 ∈ V)
5	4, 4	xpexd 7476	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 × 𝑋) ∈ V)
6		simpr2 1191	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ⊆ (𝑋 × 𝑋))
7	5, 6	ssexd 5230	. . 3 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ∈ V)
8	4, 7	jca 514	. 2 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
9		sseq1 3994	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
10		xpeq12 5582	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
11	10	anidms 569	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 × 𝑥) = (𝑋 × 𝑋))
12	11	sseq2d 4001	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑋 × 𝑋)))
13		weeq2 5546	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑋))
14	9, 12, 13	3anbi123d 1432	. . . . 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 7165	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥𝐹𝑟) = (𝑋𝐹𝑟))
17	16	eleq1d 2899	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑟) ∈ 𝐴))
18	15, 17	imbi12d 347	. . 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 3994	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 ⊆ (𝑋 × 𝑋) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
20		weeq1 5545	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 We 𝑋 ↔ 𝑅 We 𝑋))
21	19, 20	3anbi23d 1435	. . . . 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 7166	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑋𝐹𝑟) = (𝑋𝐹𝑅))
24	23	eleq1d 2899	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑋𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑅) ∈ 𝐴))
25	22, 24	imbi12d 347	. . 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 3574	. 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 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  Vcvv 3496  [wsbc 3774   ∩ cin 3937   ⊆ wss 3938  {csn 4569  {copab 5130   We wwe 5515   × cxp 5555  ^◡ccnv 5556   “ cima 5560  (class class class)co 7158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-iota 6316  df-fv 6365  df-ov 7161

This theorem is referenced by:  fpwwe2lem13  10066