Metamath Proof Explorer

Theorem elovmpt3rab

Description: Implications for the value of an operation defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by AV, 17-Jul-2018) (Revised by AV, 16-May-2019)

		Ref	Expression
	Hypothesis	elovmpt3rab.o	$⊢ O = (x \in V, y \in V ⟼ (z \in M ⟼ \{a \in N \| φ\}))$
	Assertion	elovmpt3rab	$⊢ (M \in U \land N \in T) \to (A \in (X O Y) (Z) \to ((X \in V \land Y \in V) \land (Z \in M \land A \in N)))$

Proof

Step	Hyp	Ref	Expression
1		elovmpt3rab.o	$⊢ O = (x \in V, y \in V ⟼ (z \in M ⟼ \{a \in N \| φ\}))$
2		eqidd	$⊢ (x = X \land y = Y) \to M = M$
3		eqidd	$⊢ (x = X \land y = Y) \to N = N$
4	1 2 3	elovmpt3rab1	$⊢ (M \in U \land N \in T) \to (A \in (X O Y) (Z) \to ((X \in V \land Y \in V) \land (Z \in M \land A \in N)))$