Theorem tposideq2

Description: Two ways of expressing the swap function. (Contributed by Zhi Wang, 6-Oct-2025)

	Ref	Expression
Hypothesis	tposideq2.1	$$⊢R=A\times B$$
Assertion	tposideq2	$$⊢{tpos\mathrm{I}\upharpoonright }_R=\left(x\in R⟼\bigcup {\left\{x\right\}}^{-1}\right)$$

Proof

Step	Hyp	Ref	Expression
1		tposideq2.1	$$⊢R=A\times B$$
2		relxp	$$⊢\mathrm{Rel}\left(A\times B\right)$$
3	1	releqi	$$⊢\mathrm{Rel}R\leftrightarrow \mathrm{Rel}\left(A\times B\right)$$
4	2 3	mpbir	$$⊢\mathrm{Rel}R$$
5		tposideq	$$⊢\mathrm{Rel}R\to {tpos\mathrm{I}\upharpoonright }_R=\left(x\in R⟼\bigcup {\left\{x\right\}}^{-1}\right)$$
6	4 5	ax-mp	$$⊢{tpos\mathrm{I}\upharpoonright }_R=\left(x\in R⟼\bigcup {\left\{x\right\}}^{-1}\right)$$