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Metamath Proof Explorer

Theorem funimaex

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep . Axiom 39(vi) of Quine p. 284. Compare Exercise 9 of TakeutiZaring p. 29. (Contributed by NM, 17-Nov-2002)

		Ref	Expression
	Hypothesis	zfrep5.1	⊢ 𝐵 ∈ V
	Assertion	funimaex	⊢ ( Fun 𝐴 → ( 𝐴 “ 𝐵 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		zfrep5.1	⊢ 𝐵 ∈ V
2		funimaexg	⊢ ( ( Fun 𝐴 ∧ 𝐵 ∈ V ) → ( 𝐴 “ 𝐵 ) ∈ V )
3	1 2	mpan2	⊢ ( Fun 𝐴 → ( 𝐴 “ 𝐵 ) ∈ V )