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Metamath Proof Explorer

Theorem isoeq4

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

Ref Expression
Assertion isoeq4 ( 𝐴 = 𝐶 → ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐻 Isom 𝑅 , 𝑆 ( 𝐶 , 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 f1oeq2 ⊢ ( 𝐴 = 𝐶 → ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐻 : 𝐶 –1-1-onto→ 𝐵 ) )
2 raleq ⊢ ( 𝐴 = 𝐶 → ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ 𝐶 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) )
3 2 raleqbi1dv ⊢ ( 𝐴 = 𝐶 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝐶 ∀ 𝑦 ∈ 𝐶 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) )
4 1 3 anbi12d ⊢ ( 𝐴 = 𝐶 → ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) ↔ ( 𝐻 : 𝐶 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐶 ∀ 𝑦 ∈ 𝐶 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) ) )
5 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 ) ) ) )
6 df-isom ⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐶 , 𝐵 ) ↔ ( 𝐻 : 𝐶 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐶 ∀ 𝑦 ∈ 𝐶 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) )
7 4 5 6 3bitr4g ⊢ ( 𝐴 = 𝐶 → ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐻 Isom 𝑅 , 𝑆 ( 𝐶 , 𝐵 ) ) )