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Metamath Proof Explorer

Theorem funrnex

Description: If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of Monk1 p. 46. This theorem is derived using the Axiom of Replacement in the form of funex . (Contributed by NM, 11-Nov-1995)

		Ref	Expression
	Assertion	funrnex	⊢ ( dom 𝐹 ∈ 𝐵 → ( Fun 𝐹 → ran 𝐹 ∈ V ) )

Proof

Step	Hyp	Ref	Expression
1		funex	⊢ ( ( Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ) → 𝐹 ∈ V )
2	1	ex	⊢ ( Fun 𝐹 → ( dom 𝐹 ∈ 𝐵 → 𝐹 ∈ V ) )
3		rnexg	⊢ ( 𝐹 ∈ V → ran 𝐹 ∈ V )
4	2 3	syl6com	⊢ ( dom 𝐹 ∈ 𝐵 → ( Fun 𝐹 → ran 𝐹 ∈ V ) )