Metamath Proof Explorer

Theorem ineccnvmo2

Description: Equivalence of a double universal quantification restricted to the range and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 4-Sep-2021)

		Ref	Expression
	Assertion	ineccnvmo2	$⊢ \forall x \in ran F \forall y \in ran F (x = y \lor [x] F^{-1} \cap [y] F^{-1} = \emptyset) \leftrightarrow \forall u \exists^{*} x u F x$

Proof

Step	Hyp	Ref	Expression
1		ineccnvmo	$⊢ \forall x \in ran F \forall y \in ran F (x = y \lor [x] F^{-1} \cap [y] F^{-1} = \emptyset) \leftrightarrow \forall u \exists^{*} x \in ran F u F x$
2		alrmomorn	$⊢ \forall u \exists^{} x \in ran F u F x \leftrightarrow \forall u \exists^{} x u F x$
3	1 2	bitri	$⊢ \forall x \in ran F \forall y \in ran F (x = y \lor [x] F^{-1} \cap [y] F^{-1} = \emptyset) \leftrightarrow \forall u \exists^{*} x u F x$