Theorem elima

Description: Membership in an image. Theorem 34 of Suppes p. 65. (Contributed by NM, 19-Apr-2004)

	Ref	Expression
Hypothesis	elima.1	⊢ 𝐴 ∈ V
Assertion	elima	⊢ ( 𝐴 ∈ ( 𝐵 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 𝑥 𝐵 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		elima.1	⊢ 𝐴 ∈ V
2		elimag	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ ( 𝐵 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 𝑥 𝐵 𝐴 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ ( 𝐵 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 𝑥 𝐵 𝐴 )