Metamath Proof Explorer

Theorem rbropap

Description: Properties of a pair in a restricted binary relation M expressed as an ordered-pair class abstraction: M is the binary relation W restricted by the condition ps . (Contributed by AV, 31-Jan-2021)

	Ref	Expression
Hypotheses	rbropapd.1	$⊢ φ \to M = \{⟨f, p⟩ \| (f W p \land ψ)\}$
	rbropapd.2	$⊢ (f = F \land p = P) \to (ψ \leftrightarrow χ)$
Assertion	rbropap	$⊢ (φ \land F \in X \land P \in Y) \to (F M P \leftrightarrow (F W P \land χ))$

Proof

Step	Hyp	Ref	Expression
1		rbropapd.1	$⊢ φ \to M = \{⟨f, p⟩ \| (f W p \land ψ)\}$
2		rbropapd.2	$⊢ (f = F \land p = P) \to (ψ \leftrightarrow χ)$
3	1 2	rbropapd	$⊢ φ \to ((F \in X \land P \in Y) \to (F M P \leftrightarrow (F W P \land χ)))$
4	3	3impib	$⊢ (φ \land F \in X \land P \in Y) \to (F M P \leftrightarrow (F W P \land χ))$