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Metamath Proof Explorer

Theorem rnsnopg

Description: The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004) (Revised by Mario Carneiro, 30-Apr-2015)

		Ref	Expression
	Assertion	rnsnopg	$⊢ A \in V \to ran \{⟨A, B⟩\} = \{B\}$

Proof

Step	Hyp	Ref	Expression
1		df-rn	$⊢ ran \{⟨A, B⟩\} = dom {\{⟨A, B⟩\}}^{-1}$
2		dfdm4	$⊢ dom \{⟨B, A⟩\} = ran {\{⟨B, A⟩\}}^{-1}$
3		df-rn	$⊢ ran {\{⟨B, A⟩\}}^{-1} = dom {\{⟨B, A⟩\}}^{-1}^{-1}$
4		cnvcnvsn	$⊢ {\{⟨B, A⟩\}}^{-1}^{-1} = {\{⟨A, B⟩\}}^{-1}$
5	4	dmeqi	$⊢ dom {\{⟨B, A⟩\}}^{-1}^{-1} = dom {\{⟨A, B⟩\}}^{-1}$
6	2 3 5	3eqtri	$⊢ dom \{⟨B, A⟩\} = dom {\{⟨A, B⟩\}}^{-1}$
7	1 6	eqtr4i	$⊢ ran \{⟨A, B⟩\} = dom \{⟨B, A⟩\}$
8		dmsnopg	$⊢ A \in V \to dom \{⟨B, A⟩\} = \{B\}$
9	7 8	eqtrid	$⊢ A \in V \to ran \{⟨A, B⟩\} = \{B\}$