Theorem elima2

Description: Membership in an image. Theorem 34 of Suppes p. 65. (Contributed by NM, 11-Aug-2004)

	Ref	Expression
Hypothesis	elima.1	⊢ 𝐴 ∈ V
Assertion	elima2	⊢ ( 𝐴 ∈ ( 𝐵 “ 𝐶 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐵 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		elima.1	⊢ 𝐴 ∈ V
2	1	elima	⊢ ( 𝐴 ∈ ( 𝐵 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 𝑥 𝐵 𝐴 )
3		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐶 𝑥 𝐵 𝐴 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐵 𝐴 ) )
4	2 3	bitri	⊢ ( 𝐴 ∈ ( 𝐵 “ 𝐶 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐵 𝐴 ) )