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Metamath Proof Explorer

Theorem isose

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

Ref Expression
Assertion isose ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵 ) )

Proof

Step Hyp Ref Expression
1 id ⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) )
2 isof1o ⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → 𝐻 : 𝐴 –1-1-onto→ 𝐵 )
3 f1ofun ⊢ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 → Fun 𝐻 )
4 vex ⊢ 𝑥 ∈ V
5 4 funimaex ⊢ ( Fun 𝐻 → ( 𝐻 “ 𝑥 ) ∈ V )
6 2 3 5 3syl ⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝐻 “ 𝑥 ) ∈ V )
7 1 6 isoselem ⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 Se 𝐴 → 𝑆 Se 𝐵 ) )
8 isocnv ⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ◡ 𝐻 Isom 𝑆 , 𝑅 ( 𝐵 , 𝐴 ) )
9 isof1o ⊢ ( ◡ 𝐻 Isom 𝑆 , 𝑅 ( 𝐵 , 𝐴 ) → ◡ 𝐻 : 𝐵 –1-1-onto→ 𝐴 )
10 f1ofun ⊢ ( ◡ 𝐻 : 𝐵 –1-1-onto→ 𝐴 → Fun ◡ 𝐻 )
11 4 funimaex ⊢ ( Fun ◡ 𝐻 → ( ◡ 𝐻 “ 𝑥 ) ∈ V )
12 8 9 10 11 4syl ⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( ◡ 𝐻 “ 𝑥 ) ∈ V )
13 8 12 isoselem ⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑆 Se 𝐵 → 𝑅 Se 𝐴 ) )
14 7 13 impbid ⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵 ) )