Metamath Proof Explorer


Theorem trcleq2lem

Description: Equality implies bijection. (Contributed by RP, 5-May-2020)

Ref Expression
Assertion trcleq2lem ( 𝐴 = 𝐵 → ( ( 𝑅𝐴 ∧ ( 𝐴𝐴 ) ⊆ 𝐴 ) ↔ ( 𝑅𝐵 ∧ ( 𝐵𝐵 ) ⊆ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 sseq2 ( 𝐴 = 𝐵 → ( 𝑅𝐴𝑅𝐵 ) )
2 id ( 𝐴 = 𝐵𝐴 = 𝐵 )
3 2 2 coeq12d ( 𝐴 = 𝐵 → ( 𝐴𝐴 ) = ( 𝐵𝐵 ) )
4 3 2 sseq12d ( 𝐴 = 𝐵 → ( ( 𝐴𝐴 ) ⊆ 𝐴 ↔ ( 𝐵𝐵 ) ⊆ 𝐵 ) )
5 1 4 anbi12d ( 𝐴 = 𝐵 → ( ( 𝑅𝐴 ∧ ( 𝐴𝐴 ) ⊆ 𝐴 ) ↔ ( 𝑅𝐵 ∧ ( 𝐵𝐵 ) ⊆ 𝐵 ) ) )