Description: Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | ereq2 | ⊢ ( 𝐴 = 𝐵 → ( 𝑅 Er 𝐴 ↔ 𝑅 Er 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 | ⊢ ( 𝐴 = 𝐵 → ( dom 𝑅 = 𝐴 ↔ dom 𝑅 = 𝐵 ) ) | |
2 | 1 | 3anbi2d | ⊢ ( 𝐴 = 𝐵 → ( ( Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ ( ◡ 𝑅 ∪ ( 𝑅 ∘ 𝑅 ) ) ⊆ 𝑅 ) ↔ ( Rel 𝑅 ∧ dom 𝑅 = 𝐵 ∧ ( ◡ 𝑅 ∪ ( 𝑅 ∘ 𝑅 ) ) ⊆ 𝑅 ) ) ) |
3 | df-er | ⊢ ( 𝑅 Er 𝐴 ↔ ( Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ ( ◡ 𝑅 ∪ ( 𝑅 ∘ 𝑅 ) ) ⊆ 𝑅 ) ) | |
4 | df-er | ⊢ ( 𝑅 Er 𝐵 ↔ ( Rel 𝑅 ∧ dom 𝑅 = 𝐵 ∧ ( ◡ 𝑅 ∪ ( 𝑅 ∘ 𝑅 ) ) ⊆ 𝑅 ) ) | |
5 | 2 3 4 | 3bitr4g | ⊢ ( 𝐴 = 𝐵 → ( 𝑅 Er 𝐴 ↔ 𝑅 Er 𝐵 ) ) |