Theorem isofr2

Description: A weak form of isofr that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014)

	Ref	Expression
Assertion	isofr2	$$⊢\left(H{Isom}_{R,S}\left(A,B\right)\wedge B\in V\right)\to \left(S\mathrm{Fr}B\to R\mathrm{Fr}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(H{Isom}_{R,S}\left(A,B\right)\wedge B\in V\right)\to H{Isom}_{R,S}\left(A,B\right)$$
2		imassrn	$$⊢H\left[x\right]\subseteq \mathrm{ran}H$$
3		isof1o	$$⊢H{Isom}_{R,S}\left(A,B\right)\to H:A\underset{1-1\; onto}{⟶}B$$
4		f1of	$$⊢H:A\underset{1-1\; onto}{⟶}B\to H:A⟶B$$
5		frn	$$⊢H:A⟶B\to \mathrm{ran}H\subseteq B$$
6	3 4 5	3syl	$$⊢H{Isom}_{R,S}\left(A,B\right)\to \mathrm{ran}H\subseteq B$$
7	2 6	sstrid	$$⊢H{Isom}_{R,S}\left(A,B\right)\to H\left[x\right]\subseteq B$$
8		ssexg	$$⊢\left(H\left[x\right]\subseteq B\wedge B\in V\right)\to H\left[x\right]\in \mathrm{V}$$
9	7 8	sylan	$$⊢\left(H{Isom}_{R,S}\left(A,B\right)\wedge B\in V\right)\to H\left[x\right]\in \mathrm{V}$$
10	1 9	isofrlem	$$⊢\left(H{Isom}_{R,S}\left(A,B\right)\wedge B\in V\right)\to \left(S\mathrm{Fr}B\to R\mathrm{Fr}A\right)$$