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Theorem trcleq12lem 13529
Description: Equality implies bijection. (Contributed by RP, 9-May-2020.)
Assertion
Ref Expression
trcleq12lem ((𝑅 = 𝑆𝐴 = 𝐵) → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

Proof of Theorem trcleq12lem
StepHypRef Expression
1 cleq1lem 13518 . 2 (𝑅 = 𝑆 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐴 ∧ (𝐴𝐴) ⊆ 𝐴)))
2 trcleq2lem 13527 . 2 (𝐴 = 𝐵 → ((𝑆𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
31, 2sylan9bb 732 1 ((𝑅 = 𝑆𝐴 = 𝐵) → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wss 3540  ccom 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-in 3547  df-ss 3554  df-br 4579  df-opab 4639  df-co 5037
This theorem is referenced by: (None)
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