Description: Equality implies bijection. (Contributed by RP, 5-May-2020)
Ref | Expression | ||
---|---|---|---|
Assertion | trcleq2lem | ⊢ ( 𝐴 = 𝐵 → ( ( 𝑅 ⊆ 𝐴 ∧ ( 𝐴 ∘ 𝐴 ) ⊆ 𝐴 ) ↔ ( 𝑅 ⊆ 𝐵 ∧ ( 𝐵 ∘ 𝐵 ) ⊆ 𝐵 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 | ⊢ ( 𝐴 = 𝐵 → ( 𝑅 ⊆ 𝐴 ↔ 𝑅 ⊆ 𝐵 ) ) | |
2 | id | ⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 ) | |
3 | 2 2 | coeq12d | ⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∘ 𝐴 ) = ( 𝐵 ∘ 𝐵 ) ) |
4 | 3 2 | sseq12d | ⊢ ( 𝐴 = 𝐵 → ( ( 𝐴 ∘ 𝐴 ) ⊆ 𝐴 ↔ ( 𝐵 ∘ 𝐵 ) ⊆ 𝐵 ) ) |
5 | 1 4 | anbi12d | ⊢ ( 𝐴 = 𝐵 → ( ( 𝑅 ⊆ 𝐴 ∧ ( 𝐴 ∘ 𝐴 ) ⊆ 𝐴 ) ↔ ( 𝑅 ⊆ 𝐵 ∧ ( 𝐵 ∘ 𝐵 ) ⊆ 𝐵 ) ) ) |