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Metamath Proof Explorer

Theorem funeq

Description: Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994)

		Ref	Expression
	Assertion	funeq	$⊢ A = B \to (Fun A \leftrightarrow Fun B)$

Step	Hyp	Ref	Expression
1		eqimss2	$⊢ A = B \to B \subseteq A$
2		funss	$⊢ B \subseteq A \to (Fun A \to Fun B)$
3	1 2	syl	$⊢ A = B \to (Fun A \to Fun B)$
4		eqimss	$⊢ A = B \to A \subseteq B$
5		funss	$⊢ A \subseteq B \to (Fun B \to Fun A)$
6	4 5	syl	$⊢ A = B \to (Fun B \to Fun A)$
7	3 6	impbid	$⊢ A = B \to (Fun A \leftrightarrow Fun B)$