Theorem fpwwe2lem5 9456

Description: Lemma for fpwwe2 9465. (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 481	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝐴 ∈ V)
3		simpr1 1067	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋 ⊆ 𝐴)
4	2, 3	ssexd 4805	. . 3 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋 ∈ V)
5		xpexg 6960	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
6	4, 4, 5	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 × 𝑋) ∈ V)
7		simpr2 1068	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ⊆ (𝑋 × 𝑋))
8	6, 7	ssexd 4805	. . 3 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ∈ V)
9	4, 8	jca 554	. 2 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
10		sseq1 3626	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
11		xpeq12 5134	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
12	11	anidms 677	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 × 𝑥) = (𝑋 × 𝑋))
13	12	sseq2d 3633	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑋 × 𝑋)))
14		weeq2 5103	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑋))
15	10, 13, 14	3anbi123d 1399	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)))
16	15	anbi2d 740	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) ↔ (𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋))))
17		oveq1 6657	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥𝐹𝑟) = (𝑋𝐹𝑟))
18	17	eleq1d 2686	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑟) ∈ 𝐴))
19	16, 18	imbi12d 334	. . 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 3626	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 ⊆ (𝑋 × 𝑋) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
21		weeq1 5102	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 We 𝑋 ↔ 𝑅 We 𝑋))
22	20, 21	3anbi23d 1402	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)))
23	22	anbi2d 740	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)) ↔ (𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋))))
24		oveq2 6658	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑋𝐹𝑟) = (𝑋𝐹𝑅))
25	24	eleq1d 2686	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑋𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑅) ∈ 𝐴))
26	23, 25	imbi12d 334	. . 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 3270	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑅 ∈ V) → ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴))
29	9, 28	mpcom 38	1 ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  [wsbc 3435   ∩ cin 3573   ⊆ wss 3574  {csn 4177  {copab 4712   We wwe 5072   × cxp 5112  ^◡ccnv 5113   “ cima 5117  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  fpwwe2lem13  9464