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Theorem trcleq1 14883
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 14877 1 (𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} = {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  {cab 2710  wss 3914   cint 4911  ccom 5641
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3449  df-in 3921  df-ss 3931  df-int 4912
This theorem is referenced by:  trclfv  14894
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