Theorem fpwwecbv

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

	Ref	Expression
Hypothesis	fpwwe.1	⊢ 𝑊 = { ⟨ 𝑥 , 𝑟 ⟩ ∣ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ) ) }
Assertion	fpwwecbv	⊢ 𝑊 = { ⟨ 𝑎 , 𝑠 ⟩ ∣ ( ( 𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ ( 𝑎 × 𝑎 ) ) ∧ ( 𝑠 We 𝑎 ∧ ∀ 𝑧 ∈ 𝑎 ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) ) }

Proof

Step	Hyp	Ref	Expression
1		fpwwe.1	⊢ 𝑊 = { ⟨ 𝑥 , 𝑟 ⟩ ∣ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ) ) }
2		simpl	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → 𝑥 = 𝑎 )
3	2	sseq1d	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( 𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴 ) )
4		simpr	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → 𝑟 = 𝑠 )
5	2	sqxpeqd	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( 𝑥 × 𝑥 ) = ( 𝑎 × 𝑎 ) )
6	4 5	sseq12d	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( 𝑟 ⊆ ( 𝑥 × 𝑥 ) ↔ 𝑠 ⊆ ( 𝑎 × 𝑎 ) ) )
7	3 6	anbi12d	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ↔ ( 𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ ( 𝑎 × 𝑎 ) ) ) )
8		weeq2	⊢ ( 𝑥 = 𝑎 → ( 𝑟 We 𝑥 ↔ 𝑟 We 𝑎 ) )
9		weeq1	⊢ ( 𝑟 = 𝑠 → ( 𝑟 We 𝑎 ↔ 𝑠 We 𝑎 ) )
10	8 9	sylan9bb	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( 𝑟 We 𝑥 ↔ 𝑠 We 𝑎 ) )
11		sneq	⊢ ( 𝑦 = 𝑧 → { 𝑦 } = { 𝑧 } )
12	11	imaeq2d	⊢ ( 𝑦 = 𝑧 → ( ◡ 𝑟 “ { 𝑦 } ) = ( ◡ 𝑟 “ { 𝑧 } ) )
13	12	fveq2d	⊢ ( 𝑦 = 𝑧 → ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑧 } ) ) )
14		id	⊢ ( 𝑦 = 𝑧 → 𝑦 = 𝑧 )
15	13 14	eqeq12d	⊢ ( 𝑦 = 𝑧 → ( ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ↔ ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑧 } ) ) = 𝑧 ) )
16	15	cbvralvw	⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ↔ ∀ 𝑧 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑧 } ) ) = 𝑧 )
17	4	cnveqd	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ◡ 𝑟 = ◡ 𝑠 )
18	17	imaeq1d	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( ◡ 𝑟 “ { 𝑧 } ) = ( ◡ 𝑠 “ { 𝑧 } ) )
19	18	fveqeq2d	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑧 } ) ) = 𝑧 ↔ ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) )
20	2 19	raleqbidv	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( ∀ 𝑧 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑧 } ) ) = 𝑧 ↔ ∀ 𝑧 ∈ 𝑎 ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) )
21	16 20	syl5bb	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ↔ ∀ 𝑧 ∈ 𝑎 ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) )
22	10 21	anbi12d	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ) ↔ ( 𝑠 We 𝑎 ∧ ∀ 𝑧 ∈ 𝑎 ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) ) )
23	7 22	anbi12d	⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ) ) ↔ ( ( 𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ ( 𝑎 × 𝑎 ) ) ∧ ( 𝑠 We 𝑎 ∧ ∀ 𝑧 ∈ 𝑎 ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) ) ) )
24	23	cbvopabv	⊢ { ⟨ 𝑥 , 𝑟 ⟩ ∣ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑥 × 𝑥 ) ) ∧ ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ) ) } = { ⟨ 𝑎 , 𝑠 ⟩ ∣ ( ( 𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ ( 𝑎 × 𝑎 ) ) ∧ ( 𝑠 We 𝑎 ∧ ∀ 𝑧 ∈ 𝑎 ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) ) }
25	1 24	eqtri	⊢ 𝑊 = { ⟨ 𝑎 , 𝑠 ⟩ ∣ ( ( 𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ ( 𝑎 × 𝑎 ) ) ∧ ( 𝑠 We 𝑎 ∧ ∀ 𝑧 ∈ 𝑎 ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) ) }