Theorem isofr

Description: An isomorphism preserves well-foundedness. Proposition 6.32(1) of TakeutiZaring p. 33. (Contributed by NM, 30-Apr-2004) (Revised by Mario Carneiro, 18-Nov-2014)

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

Proof

Step	Hyp	Ref	Expression
1		isocnv	$$⊢H{Isom}_{R,S}\left(A,B\right)\to H^{-1}{Isom}_{S,R}\left(B,A\right)$$
2		id	$$⊢H^{-1}{Isom}_{S,R}\left(B,A\right)\to H^{-1}{Isom}_{S,R}\left(B,A\right)$$
3		isof1o	$$⊢H^{-1}{Isom}_{S,R}\left(B,A\right)\to H^{-1}:B\underset{1-1\; onto}{⟶}A$$
4		f1ofun	$$⊢H^{-1}:B\underset{1-1\; onto}{⟶}A\to \mathrm{Fun}{H}^{-1}$$
5		vex	$$⊢x\in \mathrm{V}$$
6	5	funimaex	$$⊢\mathrm{Fun}{H}^{-1}\to H^{-1}\left[x\right]\in \mathrm{V}$$
7	3 4 6	3syl	$$⊢H^{-1}{Isom}_{S,R}\left(B,A\right)\to H^{-1}\left[x\right]\in \mathrm{V}$$
8	2 7	isofrlem	$$⊢H^{-1}{Isom}_{S,R}\left(B,A\right)\to \left(R\mathrm{Fr}A\to S\mathrm{Fr}B\right)$$
9	1 8	syl	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \left(R\mathrm{Fr}A\to S\mathrm{Fr}B\right)$$
10		id	$$⊢H{Isom}_{R,S}\left(A,B\right)\to H{Isom}_{R,S}\left(A,B\right)$$
11		isof1o	$$⊢H{Isom}_{R,S}\left(A,B\right)\to H:A\underset{1-1\; onto}{⟶}B$$
12		f1ofun	$$⊢H:A\underset{1-1\; onto}{⟶}B\to \mathrm{Fun}H$$
13	5	funimaex	$$⊢\mathrm{Fun}H\to H\left[x\right]\in \mathrm{V}$$
14	11 12 13	3syl	$$⊢H{Isom}_{R,S}\left(A,B\right)\to H\left[x\right]\in \mathrm{V}$$
15	10 14	isofrlem	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \left(S\mathrm{Fr}B\to R\mathrm{Fr}A\right)$$
16	9 15	impbid	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \left(R\mathrm{Fr}A\leftrightarrow S\mathrm{Fr}B\right)$$