Theorem isofr2

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

	Ref	Expression
Assertion	isofr2	⊢ ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ 𝐵 ∈ 𝑉 ) → ( 𝑆 Fr 𝐵 → 𝑅 Fr 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ 𝐵 ∈ 𝑉 ) → 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) )
2		imassrn	⊢ ( 𝐻 “ 𝑥 ) ⊆ ran 𝐻
3		isof1o	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → 𝐻 : 𝐴 –1-1-onto→ 𝐵 )
4		f1of	⊢ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 → 𝐻 : 𝐴 ⟶ 𝐵 )
5		frn	⊢ ( 𝐻 : 𝐴 ⟶ 𝐵 → ran 𝐻 ⊆ 𝐵 )
6	3 4 5	3syl	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ran 𝐻 ⊆ 𝐵 )
7	2 6	sstrid	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝐻 “ 𝑥 ) ⊆ 𝐵 )
8		ssexg	⊢ ( ( ( 𝐻 “ 𝑥 ) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ) → ( 𝐻 “ 𝑥 ) ∈ V )
9	7 8	sylan	⊢ ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ 𝐵 ∈ 𝑉 ) → ( 𝐻 “ 𝑥 ) ∈ V )
10	1 9	isofrlem	⊢ ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ 𝐵 ∈ 𝑉 ) → ( 𝑆 Fr 𝐵 → 𝑅 Fr 𝐴 ) )