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Metamath Proof Explorer

Theorem isoeq1

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

Ref Expression
Assertion isoeq1 ( 𝐻 = 𝐺 → ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐺 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 f1oeq1 ⊢ ( 𝐻 = 𝐺 → ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐺 : 𝐴 –1-1-onto→ 𝐵 ) )
2 fveq1 ⊢ ( 𝐻 = 𝐺 → ( 𝐻 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) )
3 fveq1 ⊢ ( 𝐻 = 𝐺 → ( 𝐻 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) )
4 2 3 breq12d ⊢ ( 𝐻 = 𝐺 → ( ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ↔ ( 𝐺 ‘ 𝑥 ) 𝑆 ( 𝐺 ‘ 𝑦 ) ) )
5 4 bibi2d ⊢ ( 𝐻 = 𝐺 → ( ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ( 𝑥 𝑅 𝑦 ↔ ( 𝐺 ‘ 𝑥 ) 𝑆 ( 𝐺 ‘ 𝑦 ) ) ) )
6 5 2ralbidv ⊢ ( 𝐻 = 𝐺 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐺 ‘ 𝑥 ) 𝑆 ( 𝐺 ‘ 𝑦 ) ) ) )
7 1 6 anbi12d ⊢ ( 𝐻 = 𝐺 → ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) ↔ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐺 ‘ 𝑥 ) 𝑆 ( 𝐺 ‘ 𝑦 ) ) ) ) )
8 df-isom ⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) )
9 df-isom ⊢ ( 𝐺 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐺 ‘ 𝑥 ) 𝑆 ( 𝐺 ‘ 𝑦 ) ) ) )
10 7 8 9 3bitr4g ⊢ ( 𝐻 = 𝐺 → ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐺 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ) )