Theorem elimag

Description: Membership in an image. Theorem 34 of Suppes p. 65. (Contributed by NM, 20-Jan-2007)

	Ref	Expression
Assertion	elimag	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ( 𝐵 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 𝑥 𝐵 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		breq2	⊢ ( 𝑦 = 𝐴 → ( 𝑥 𝐵 𝑦 ↔ 𝑥 𝐵 𝐴 ) )
2	1	rexbidv	⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑥 ∈ 𝐶 𝑥 𝐵 𝑦 ↔ ∃ 𝑥 ∈ 𝐶 𝑥 𝐵 𝐴 ) )
3		dfima2	⊢ ( 𝐵 “ 𝐶 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐶 𝑥 𝐵 𝑦 }
4	2 3	elab2g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ( 𝐵 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 𝑥 𝐵 𝐴 ) )