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Metamath Proof Explorer

Theorem isorel

Description: An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004)

Ref Expression
Assertion isorel ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( 𝐶 𝑅 𝐷 ↔ ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝐷 ) ) )

Proof

Step Hyp Ref Expression
1 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 ) ) ) )
2 1 simprbi ⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) )
3 breq1 ⊢ ( 𝑥 = 𝐶 → ( 𝑥 𝑅 𝑦 ↔ 𝐶 𝑅 𝑦 ) )
4 fveq2 ⊢ ( 𝑥 = 𝐶 → ( 𝐻 ‘ 𝑥 ) = ( 𝐻 ‘ 𝐶 ) )
5 4 breq1d ⊢ ( 𝑥 = 𝐶 → ( ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ↔ ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) )
6 3 5 bibi12d ⊢ ( 𝑥 = 𝐶 → ( ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ( 𝐶 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) )
7 breq2 ⊢ ( 𝑦 = 𝐷 → ( 𝐶 𝑅 𝑦 ↔ 𝐶 𝑅 𝐷 ) )
8 fveq2 ⊢ ( 𝑦 = 𝐷 → ( 𝐻 ‘ 𝑦 ) = ( 𝐻 ‘ 𝐷 ) )
9 8 breq2d ⊢ ( 𝑦 = 𝐷 → ( ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ↔ ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝐷 ) ) )
10 7 9 bibi12d ⊢ ( 𝑦 = 𝐷 → ( ( 𝐶 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ( 𝐶 𝑅 𝐷 ↔ ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝐷 ) ) ) )
11 6 10 rspc2v ⊢ ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) → ( 𝐶 𝑅 𝐷 ↔ ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝐷 ) ) ) )
12 2 11 mpan9 ⊢ ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( 𝐶 𝑅 𝐷 ↔ ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝐷 ) ) )