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Metamath Proof Explorer

Theorem isoeq5

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

Ref Expression
Assertion isoeq5 ( 𝐵 = 𝐶 → ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 f1oeq3 ⊢ ( 𝐵 = 𝐶 → ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐻 : 𝐴 –1-1-onto→ 𝐶 ) )
2 1 anbi1d ⊢ ( 𝐵 = 𝐶 → ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) ↔ ( 𝐻 : 𝐴 –1-1-onto→ 𝐶 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) ) )
3 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 ) ) ) )
4 df-isom ⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐶 ) ↔ ( 𝐻 : 𝐴 –1-1-onto→ 𝐶 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) )
5 2 3 4 3bitr4g ⊢ ( 𝐵 = 𝐶 → ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐶 ) ) )