Theorem tposidf1o

Description: The swap function, or the twisting map, is bijective. (Contributed by Zhi Wang, 5-Oct-2025)

	Ref	Expression
Assertion	tposidf1o	$$⊢tpos\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right):B\times A\underset{1-1\; onto}{⟶}A\times B$$

Proof

Step	Hyp	Ref	Expression
1		f1oi	$$⊢\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right):A\times B\underset{1-1\; onto}{⟶}A\times B$$
2		tposf1o	$$⊢\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right):A\times B\underset{1-1\; onto}{⟶}A\times B\to tpos\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right):B\times A\underset{1-1\; onto}{⟶}A\times B$$
3	1 2	ax-mp	$$⊢tpos\left({\mathrm{I}\upharpoonright }_{\left(A\times B\right)}\right):B\times A\underset{1-1\; onto}{⟶}A\times B$$