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Metamath Proof Explorer

Theorem dmfexALT

Description: Alternate proof of dmfex : shorter but using ax-rep . (Contributed by NM, 27-Aug-2006) (Proof shortened by Andrew Salmon, 17-Sep-2011) (Proof shortened by AV, 23-Aug-2024) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	dmfexALT	⊢ ( ( 𝐹 ∈ 𝐶 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → 𝐴 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐹 ∈ 𝐶 → 𝐹 ∈ V )
2		fdmexb	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝐴 ∈ V ↔ 𝐹 ∈ V ) )
3	2	biimprd	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝐹 ∈ V → 𝐴 ∈ V ) )
4	1 3	mpan9	⊢ ( ( 𝐹 ∈ 𝐶 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → 𝐴 ∈ V )