Theorem elfunsg

Description: Closed form of elfuns . (Contributed by Scott Fenton, 2-May-2014)

	Ref	Expression
Assertion	elfunsg	$$⊢F\in V\to \left(F\in \mathrm{𝖥𝗎𝗇𝗌}\leftrightarrow \mathrm{Fun}F\right)$$

Proof

Step	Hyp	Ref	Expression
1		eleq1	$$⊢f=F\to \left(f\in \mathrm{𝖥𝗎𝗇𝗌}\leftrightarrow F\in \mathrm{𝖥𝗎𝗇𝗌}\right)$$
2		funeq	$$⊢f=F\to \left(\mathrm{Fun}f\leftrightarrow \mathrm{Fun}F\right)$$
3		vex	$$⊢f\in \mathrm{V}$$
4	3	elfuns	$$⊢f\in \mathrm{𝖥𝗎𝗇𝗌}\leftrightarrow \mathrm{Fun}f$$
5	1 2 4	vtoclbg	$$⊢F\in V\to \left(F\in \mathrm{𝖥𝗎𝗇𝗌}\leftrightarrow \mathrm{Fun}F\right)$$