Theorem tposidres

Description: Swap an ordered pair. (Contributed by Zhi Wang, 5-Oct-2025)

	Ref	Expression
Hypotheses	tposidres.x	$$⊢\phi \to X\in A$$
	tposidres.y	$$⊢\phi \to Y\in B$$
Assertion	tposidres	$$⊢\phi \to Ytpos\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right)X=⟨X,Y⟩$$

Proof

Step	Hyp	Ref	Expression
1		tposidres.x	$$⊢\phi \to X\in A$$
2		tposidres.y	$$⊢\phi \to Y\in B$$
3		ovtpos	$$⊢Ytpos\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right)X=X\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right)Y$$
4		df-ov	$$⊢X\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right)Y=\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right)\left(⟨X,Y⟩\right)$$
5	3 4	eqtri	$$⊢Ytpos\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right)X=\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right)\left(⟨X,Y⟩\right)$$
6	1 2	opelxpd	$$⊢\phi \to ⟨X,Y⟩\in \left(A\times B\right)$$
7	6	fvresd	$$⊢\phi \to \left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right)\left(⟨X,Y⟩\right)=\mathrm{I}\left(⟨X,Y⟩\right)$$
8	5 7	eqtrid	$$⊢\phi \to Ytpos\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right)X=\mathrm{I}\left(⟨X,Y⟩\right)$$
9		opex	$$⊢⟨X,Y⟩\in \mathrm{V}$$
10		fvi	$$⊢⟨X,Y⟩\in \mathrm{V}\to \mathrm{I}\left(⟨X,Y⟩\right)=⟨X,Y⟩$$
11	9 10	ax-mp	$$⊢\mathrm{I}\left(⟨X,Y⟩\right)=⟨X,Y⟩$$
12	8 11	eqtrdi	$$⊢\phi \to Ytpos\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right)X=⟨X,Y⟩$$