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 ) ) )