 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  trcleq1 Structured version   Visualization version   GIF version

Theorem trcleq1 13938
 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 13932 1 (𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} = {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631  {cab 2757   ⊆ wss 3723  ∩ cint 4611   ∘ ccom 5253 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-in 3730  df-ss 3737  df-int 4612 This theorem is referenced by:  trclfv  13949
 Copyright terms: Public domain W3C validator