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Theorem trcleq12lem 14692
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 14681 . 2 (𝑅 = 𝑆 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐴 ∧ (𝐴𝐴) ⊆ 𝐴)))
2 trcleq2lem 14690 . 2 (𝐴 = 𝐵 → ((𝑆𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
31, 2sylan9bb 510 1 ((𝑅 = 𝑆𝐴 = 𝐵) → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wss 3887  ccom 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3432  df-in 3894  df-ss 3904  df-br 5075  df-opab 5137  df-co 5594
This theorem is referenced by: (None)
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