Theorem fpwwe2lem1

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

	Ref	Expression
Hypothesis	fpwwe2.1	⊢ 𝑊 = { ⟨ 𝑥 , 𝑟 ⟩ ∣ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 [ ( ◡ 𝑟 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑟 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) }
Assertion	fpwwe2lem1	⊢ 𝑊 ⊆ ( 𝒫 𝐴 × 𝒫 ( 𝐴 × 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		fpwwe2.1	⊢ 𝑊 = { ⟨ 𝑥 , 𝑟 ⟩ ∣ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 [ ( ◡ 𝑟 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑟 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) }
2		simpll	⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 [ ( ◡ 𝑟 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑟 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) → 𝑥 ⊆ 𝐴 )
3		velpw	⊢ ( 𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴 )
4	2 3	sylibr	⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 [ ( ◡ 𝑟 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑟 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) → 𝑥 ∈ 𝒫 𝐴 )
5		simplr	⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 [ ( ◡ 𝑟 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑟 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) → 𝑟 ⊆ ( 𝑥 × 𝑥 ) )
6		xpss12	⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴 ) → ( 𝑥 × 𝑥 ) ⊆ ( 𝐴 × 𝐴 ) )
7	2 2 6	syl2anc	⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 [ ( ◡ 𝑟 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑟 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) → ( 𝑥 × 𝑥 ) ⊆ ( 𝐴 × 𝐴 ) )
8	5 7	sstrd	⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 [ ( ◡ 𝑟 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑟 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) → 𝑟 ⊆ ( 𝐴 × 𝐴 ) )
9		velpw	⊢ ( 𝑟 ∈ 𝒫 ( 𝐴 × 𝐴 ) ↔ 𝑟 ⊆ ( 𝐴 × 𝐴 ) )
10	8 9	sylibr	⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 [ ( ◡ 𝑟 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑟 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) → 𝑟 ∈ 𝒫 ( 𝐴 × 𝐴 ) )
11	4 10	jca	⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 [ ( ◡ 𝑟 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑟 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) → ( 𝑥 ∈ 𝒫 𝐴 ∧ 𝑟 ∈ 𝒫 ( 𝐴 × 𝐴 ) ) )
12	11	ssopab2i	⊢ { ⟨ 𝑥 , 𝑟 ⟩ ∣ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 [ ( ◡ 𝑟 “ { 𝑦 } ) / 𝑢 ] ( 𝑢 𝐹 ( 𝑟 ∩ ( 𝑢 × 𝑢 ) ) ) = 𝑦 ) ) } ⊆ { ⟨ 𝑥 , 𝑟 ⟩ ∣ ( 𝑥 ∈ 𝒫 𝐴 ∧ 𝑟 ∈ 𝒫 ( 𝐴 × 𝐴 ) ) }
13		df-xp	⊢ ( 𝒫 𝐴 × 𝒫 ( 𝐴 × 𝐴 ) ) = { ⟨ 𝑥 , 𝑟 ⟩ ∣ ( 𝑥 ∈ 𝒫 𝐴 ∧ 𝑟 ∈ 𝒫 ( 𝐴 × 𝐴 ) ) }
14	12 1 13	3sstr4i	⊢ 𝑊 ⊆ ( 𝒫 𝐴 × 𝒫 ( 𝐴 × 𝐴 ) )