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Theorem trcleq1 14137
 Description: Equality of relations implies equality of transitive closures. (Contributed by RP, 9-May-2020.)
Assertion
Ref Expression
trcleq1 (𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} = {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
Distinct variable groups:   𝑅,𝑟   𝑆,𝑟

Proof of Theorem trcleq1
StepHypRef Expression
1 cleq1 14131 1 (𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} = {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601  {cab 2763   ⊆ wss 3792  ∩ cint 4710   ∘ ccom 5359 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-ral 3095  df-in 3799  df-ss 3806  df-int 4711 This theorem is referenced by:  trclfv  14148
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