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Metamath Proof Explorer

Theorem dffunALTV3

Description: Alternate definition of the function relation predicate, cf. dfdisjALTV3 . Reproduction of dffun2 . For the X axis and the Y axis you can convert the right side to ( A. x1 A. y1 A. y2 ( ( x1 f y1 /\ x1 f y2 ) -> y1 = y2 ) /\ Rel F ) . (Contributed by NM, 29-Dec-1996)

		Ref	Expression
	Assertion	dffunALTV3	$⊢ FunALTV F \leftrightarrow (\forall u \forall x \forall y ((u F x \land u F y) \to x = y) \land Rel F)$

Proof

Step	Hyp	Ref	Expression
1		dffunALTV2	$⊢ FunALTV F \leftrightarrow (≀ F \subseteq I \land Rel F)$
2		cossssid3	$⊢ ≀ F \subseteq I \leftrightarrow \forall u \forall x \forall y ((u F x \land u F y) \to x = y)$
3	2	anbi1i	$⊢ (≀ F \subseteq I \land Rel F) \leftrightarrow (\forall u \forall x \forall y ((u F x \land u F y) \to x = y) \land Rel F)$
4	1 3	bitri	$⊢ FunALTV F \leftrightarrow (\forall u \forall x \forall y ((u F x \land u F y) \to x = y) \land Rel F)$