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Metamath Proof Explorer

Theorem cnvopabOLD

Description: Obsolete version of cnvopab as of 7-Jun-2025. (Contributed by NM, 11-Dec-2003) (Proof shortened by Andrew Salmon, 27-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	cnvopabOLD	$⊢ {\{⟨x, y⟩ \| φ\}}^{-1} = \{⟨y, x⟩ \| φ\}$

Proof

Step	Hyp	Ref	Expression
1		relcnv	$⊢ Rel {\{⟨x, y⟩ \| φ\}}^{-1}$
2		relopabv	$⊢ Rel \{⟨y, x⟩ \| φ\}$
3		vopelopabsb	$⊢ ⟨w, z⟩ \in \{⟨x, y⟩ \| φ\} \leftrightarrow [w/ x] [z/ y] φ$
4		sbcom2	$⊢ [w/ x] [z/ y] φ \leftrightarrow [z/ y] [w/ x] φ$
5	3 4	bitri	$⊢ ⟨w, z⟩ \in \{⟨x, y⟩ \| φ\} \leftrightarrow [z/ y] [w/ x] φ$
6		vex	$⊢ z \in V$
7		vex	$⊢ w \in V$
8	6 7	opelcnv	$⊢ ⟨z, w⟩ \in {\{⟨x, y⟩ \| φ\}}^{-1} \leftrightarrow ⟨w, z⟩ \in \{⟨x, y⟩ \| φ\}$
9		vopelopabsb	$⊢ ⟨z, w⟩ \in \{⟨y, x⟩ \| φ\} \leftrightarrow [z/ y] [w/ x] φ$
10	5 8 9	3bitr4i	$⊢ ⟨z, w⟩ \in {\{⟨x, y⟩ \| φ\}}^{-1} \leftrightarrow ⟨z, w⟩ \in \{⟨y, x⟩ \| φ\}$
11	1 2 10	eqrelriiv	$⊢ {\{⟨x, y⟩ \| φ\}}^{-1} = \{⟨y, x⟩ \| φ\}$