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	$$⊢\left(F\in C\wedge F:A⟶B\right)\to A\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢F\in C\to F\in \mathrm{V}$$
2		fdmexb	$$⊢F:A⟶B\to \left(A\in \mathrm{V}\leftrightarrow F\in \mathrm{V}\right)$$
3	2	biimprd	$$⊢F:A⟶B\to \left(F\in \mathrm{V}\to A\in \mathrm{V}\right)$$
4	1 3	mpan9	$$⊢\left(F\in C\wedge F:A⟶B\right)\to A\in \mathrm{V}$$