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Theorem trcleq12lem 14329
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 14318 . 2 (𝑅 = 𝑆 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐴 ∧ (𝐴𝐴) ⊆ 𝐴)))
2 trcleq2lem 14327 . 2 (𝐴 = 𝐵 → ((𝑆𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
31, 2sylan9bb 512 1 ((𝑅 = 𝑆𝐴 = 𝐵) → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wss 3909  ccom 5531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-v 3472  df-in 3916  df-ss 3926  df-br 5039  df-opab 5101  df-co 5536
This theorem is referenced by: (None)
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