Theorem eldmg

Description: Domain membership. Theorem 4 of Suppes p. 59. (Contributed by Mario Carneiro, 9-Jul-2014)

	Ref	Expression
Assertion	eldmg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ dom 𝐵 ↔ ∃ 𝑦 𝐴 𝐵 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		breq1	⊢ ( 𝑥 = 𝑤 → ( 𝑥 𝐵 𝑦 ↔ 𝑤 𝐵 𝑦 ) )
2	1	exbidv	⊢ ( 𝑥 = 𝑤 → ( ∃ 𝑦 𝑥 𝐵 𝑦 ↔ ∃ 𝑦 𝑤 𝐵 𝑦 ) )
3		breq1	⊢ ( 𝑤 = 𝐴 → ( 𝑤 𝐵 𝑦 ↔ 𝐴 𝐵 𝑦 ) )
4	3	exbidv	⊢ ( 𝑤 = 𝐴 → ( ∃ 𝑦 𝑤 𝐵 𝑦 ↔ ∃ 𝑦 𝐴 𝐵 𝑦 ) )
5		df-dm	⊢ dom 𝐵 = { 𝑥 ∣ ∃ 𝑦 𝑥 𝐵 𝑦 }
6	2 4 5	elab2gw	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ dom 𝐵 ↔ ∃ 𝑦 𝐴 𝐵 𝑦 ) )