This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem fvpr1

Description: The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010) (Proof shortened by BJ, 26-Sep-2024)

	Ref	Expression
Hypotheses	fvpr1.1	$⊢ A \in V$
	fvpr1.2	$⊢ C \in V$
Assertion	fvpr1	$⊢ A \neq B \to \{⟨A, C⟩, ⟨B, D⟩\} (A) = C$

Proof

Step	Hyp	Ref	Expression
1		fvpr1.1	$⊢ A \in V$
2		fvpr1.2	$⊢ C \in V$
3		fvpr1g	$⊢ (A \in V \land C \in V \land A \neq B) \to \{⟨A, C⟩, ⟨B, D⟩\} (A) = C$
4	1 2 3	mp3an12	$⊢ A \neq B \to \{⟨A, C⟩, ⟨B, D⟩\} (A) = C$