Metamath Proof Explorer


Theorem isoeq3

Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004)

Ref Expression
Assertion isoeq3 ( 𝑆 = 𝑇 → ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐻 Isom 𝑅 , 𝑇 ( 𝐴 , 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 breq ( 𝑆 = 𝑇 → ( ( 𝐻𝑥 ) 𝑆 ( 𝐻𝑦 ) ↔ ( 𝐻𝑥 ) 𝑇 ( 𝐻𝑦 ) ) )
2 1 bibi2d ( 𝑆 = 𝑇 → ( ( 𝑥 𝑅 𝑦 ↔ ( 𝐻𝑥 ) 𝑆 ( 𝐻𝑦 ) ) ↔ ( 𝑥 𝑅 𝑦 ↔ ( 𝐻𝑥 ) 𝑇 ( 𝐻𝑦 ) ) ) )
3 2 2ralbidv ( 𝑆 = 𝑇 → ( ∀ 𝑥𝐴𝑦𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻𝑥 ) 𝑆 ( 𝐻𝑦 ) ) ↔ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻𝑥 ) 𝑇 ( 𝐻𝑦 ) ) ) )
4 3 anbi2d ( 𝑆 = 𝑇 → ( ( 𝐻 : 𝐴1-1-onto𝐵 ∧ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻𝑥 ) 𝑆 ( 𝐻𝑦 ) ) ) ↔ ( 𝐻 : 𝐴1-1-onto𝐵 ∧ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻𝑥 ) 𝑇 ( 𝐻𝑦 ) ) ) ) )
5 df-isom ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ ( 𝐻 : 𝐴1-1-onto𝐵 ∧ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻𝑥 ) 𝑆 ( 𝐻𝑦 ) ) ) )
6 df-isom ( 𝐻 Isom 𝑅 , 𝑇 ( 𝐴 , 𝐵 ) ↔ ( 𝐻 : 𝐴1-1-onto𝐵 ∧ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻𝑥 ) 𝑇 ( 𝐻𝑦 ) ) ) )
7 4 5 6 3bitr4g ( 𝑆 = 𝑇 → ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐻 Isom 𝑅 , 𝑇 ( 𝐴 , 𝐵 ) ) )