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Theorem fpwwe2lem1 9491
Description: Lemma for fpwwe2 9503. (Contributed by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
fpwwe2.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
Assertion
Ref Expression
fpwwe2lem1 𝑊 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))
Distinct variable groups:   𝑦,𝑢,𝑟,𝑥,𝐹   𝐴,𝑟,𝑥   𝑊,𝑟,𝑢,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑦,𝑢)

Proof of Theorem fpwwe2lem1
StepHypRef Expression
1 simpll 805 . . . . 5 (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) → 𝑥𝐴)
2 selpw 4198 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
31, 2sylibr 224 . . . 4 (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) → 𝑥 ∈ 𝒫 𝐴)
4 simplr 807 . . . . . 6 (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) → 𝑟 ⊆ (𝑥 × 𝑥))
5 xpss12 5158 . . . . . . 7 ((𝑥𝐴𝑥𝐴) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
61, 1, 5syl2anc 694 . . . . . 6 (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
74, 6sstrd 3646 . . . . 5 (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) → 𝑟 ⊆ (𝐴 × 𝐴))
8 selpw 4198 . . . . 5 (𝑟 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑟 ⊆ (𝐴 × 𝐴))
97, 8sylibr 224 . . . 4 (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
103, 9jca 553 . . 3 (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) → (𝑥 ∈ 𝒫 𝐴𝑟 ∈ 𝒫 (𝐴 × 𝐴)))
1110ssopab2i 5032 . 2 {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} ⊆ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑟 ∈ 𝒫 (𝐴 × 𝐴))}
12 fpwwe2.1 . 2 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
13 df-xp 5149 . 2 (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑟 ∈ 𝒫 (𝐴 × 𝐴))}
1411, 12, 133sstr4i 3677 1 𝑊 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wcel 2030  wral 2941  [wsbc 3468  cin 3606  wss 3607  𝒫 cpw 4191  {csn 4210  {copab 4745   We wwe 5101   × cxp 5141  ccnv 5142  cima 5146  (class class class)co 6690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193  df-opab 4746  df-xp 5149
This theorem is referenced by: (None)
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