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 ) ) )