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Theorem trcleq12lem 13941
 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 13930 . 2 (𝑅 = 𝑆 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐴 ∧ (𝐴𝐴) ⊆ 𝐴)))
2 trcleq2lem 13939 . 2 (𝐴 = 𝐵 → ((𝑆𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
31, 2sylan9bb 493 1 ((𝑅 = 𝑆𝐴 = 𝐵) → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1630   ⊆ wss 3721   ∘ ccom 5253 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-in 3728  df-ss 3735  df-br 4785  df-opab 4845  df-co 5258 This theorem is referenced by: (None)
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