Theorem isose

Description: An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015)

	Ref	Expression
Assertion	isose	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \left(R\mathrm{Se}A\leftrightarrow S\mathrm{Se}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢H{Isom}_{R,S}\left(A,B\right)\to H{Isom}_{R,S}\left(A,B\right)$$
2		isof1o	$$⊢H{Isom}_{R,S}\left(A,B\right)\to H:A\underset{1-1\; onto}{⟶}B$$
3		f1ofun	$$⊢H:A\underset{1-1\; onto}{⟶}B\to \mathrm{Fun}H$$
4		vex	$$⊢x\in \mathrm{V}$$
5	4	funimaex	$$⊢\mathrm{Fun}H\to H\left[x\right]\in \mathrm{V}$$
6	2 3 5	3syl	$$⊢H{Isom}_{R,S}\left(A,B\right)\to H\left[x\right]\in \mathrm{V}$$
7	1 6	isoselem	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \left(R\mathrm{Se}A\to S\mathrm{Se}B\right)$$
8		isocnv	$$⊢H{Isom}_{R,S}\left(A,B\right)\to H^{-1}{Isom}_{S,R}\left(B,A\right)$$
9		isof1o	$$⊢H^{-1}{Isom}_{S,R}\left(B,A\right)\to H^{-1}:B\underset{1-1\; onto}{⟶}A$$
10		f1ofun	$$⊢H^{-1}:B\underset{1-1\; onto}{⟶}A\to \mathrm{Fun}{H}^{-1}$$
11	4	funimaex	$$⊢\mathrm{Fun}{H}^{-1}\to H^{-1}\left[x\right]\in \mathrm{V}$$
12	8 9 10 11	4syl	$$⊢H{Isom}_{R,S}\left(A,B\right)\to H^{-1}\left[x\right]\in \mathrm{V}$$
13	8 12	isoselem	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \left(S\mathrm{Se}B\to R\mathrm{Se}A\right)$$
14	7 13	impbid	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \left(R\mathrm{Se}A\leftrightarrow S\mathrm{Se}B\right)$$