Theorem eltskm

Description: Belonging to ( tarskiMapA ) . (Contributed by FL, 17-Apr-2011) (Proof shortened by Mario Carneiro, 21-Sep-2014)

	Ref	Expression
Assertion	eltskm	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ↔ ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tskmval	⊢ ( 𝐴 ∈ 𝑉 → ( tarskiMap ‘ 𝐴 ) = ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } )
2	1	eleq2d	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ↔ 𝐵 ∈ ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ) )
3		elex	⊢ ( 𝐵 ∈ ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } → 𝐵 ∈ V )
4	3	a1i	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } → 𝐵 ∈ V ) )
5		tskmid	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ ( tarskiMap ‘ 𝐴 ) )
6		tskmcl	⊢ ( tarskiMap ‘ 𝐴 ) ∈ Tarski
7		eleq2	⊢ ( 𝑥 = ( tarskiMap ‘ 𝐴 ) → ( 𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ( tarskiMap ‘ 𝐴 ) ) )
8		eleq2	⊢ ( 𝑥 = ( tarskiMap ‘ 𝐴 ) → ( 𝐵 ∈ 𝑥 ↔ 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ) )
9	7 8	imbi12d	⊢ ( 𝑥 = ( tarskiMap ‘ 𝐴 ) → ( ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) ↔ ( 𝐴 ∈ ( tarskiMap ‘ 𝐴 ) → 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ) ) )
10	9	rspcv	⊢ ( ( tarskiMap ‘ 𝐴 ) ∈ Tarski → ( ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) → ( 𝐴 ∈ ( tarskiMap ‘ 𝐴 ) → 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ) ) )
11	6 10	ax-mp	⊢ ( ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) → ( 𝐴 ∈ ( tarskiMap ‘ 𝐴 ) → 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ) )
12	5 11	syl5com	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) → 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ) )
13		elex	⊢ ( 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) → 𝐵 ∈ V )
14	12 13	syl6	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) → 𝐵 ∈ V ) )
15		elintrabg	⊢ ( 𝐵 ∈ V → ( 𝐵 ∈ ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ↔ ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) ) )
16	15	a1i	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ V → ( 𝐵 ∈ ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ↔ ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) ) ) )
17	4 14 16	pm5.21ndd	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ↔ ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) ) )
18	2 17	bitrd	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ↔ ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) ) )