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Metamath Proof Explorer

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 4 2 weeq12d ⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( 𝑟 We 𝑥 ↔ 𝑠 We 𝑎 ) )
9 sneq ⊢ ( 𝑦 = 𝑧 → { 𝑦 } = { 𝑧 } )
10 9 imaeq2d ⊢ ( 𝑦 = 𝑧 → ( ◡ 𝑟 “ { 𝑦 } ) = ( ◡ 𝑟 “ { 𝑧 } ) )
11 10 fveq2d ⊢ ( 𝑦 = 𝑧 → ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑧 } ) ) )
12 id ⊢ ( 𝑦 = 𝑧 → 𝑦 = 𝑧 )
13 11 12 eqeq12d ⊢ ( 𝑦 = 𝑧 → ( ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ↔ ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑧 } ) ) = 𝑧 ) )
14 13 cbvralvw ⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ↔ ∀ 𝑧 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑧 } ) ) = 𝑧 )
15 4 cnveqd ⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ◡ 𝑟 = ◡ 𝑠 )
16 15 imaeq1d ⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( ◡ 𝑟 “ { 𝑧 } ) = ( ◡ 𝑠 “ { 𝑧 } ) )
17 16 fveqeq2d ⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑧 } ) ) = 𝑧 ↔ ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) )
18 2 17 raleqbidv ⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( ∀ 𝑧 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑧 } ) ) = 𝑧 ↔ ∀ 𝑧 ∈ 𝑎 ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) )
19 14 18 bitrid ⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ↔ ∀ 𝑧 ∈ 𝑎 ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) )
20 8 19 anbi12d ⊢ ( ( 𝑥 = 𝑎 ∧ 𝑟 = 𝑠 ) → ( ( 𝑟 We 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ ( ◡ 𝑟 “ { 𝑦 } ) ) = 𝑦 ) ↔ ( 𝑠 We 𝑎 ∧ ∀ 𝑧 ∈ 𝑎 ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) ) )
21 7 20 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 ) ) ) )
22 21 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 ) ) }
23 1 22 eqtri ⊢ 𝑊 = { ⟨ 𝑎 , 𝑠 ⟩ ∣ ( ( 𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ ( 𝑎 × 𝑎 ) ) ∧ ( 𝑠 We 𝑎 ∧ ∀ 𝑧 ∈ 𝑎 ( 𝐹 ‘ ( ◡ 𝑠 “ { 𝑧 } ) ) = 𝑧 ) ) }