 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cleq1 Structured version   Visualization version   GIF version

Theorem cleq1 13519
 Description: Equality of relations implies equality of closures. (Contributed by RP, 9-May-2020.)
Assertion
Ref Expression
cleq1 (𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
Distinct variable groups:   𝑅,𝑟   𝑆,𝑟
Allowed substitution hint:   𝜑(𝑟)

Proof of Theorem cleq1
StepHypRef Expression
1 cleq1lem 13518 . . 3 (𝑅 = 𝑆 → ((𝑅𝑟𝜑) ↔ (𝑆𝑟𝜑)))
21abbidv 2728 . 2 (𝑅 = 𝑆 → {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
32inteqd 4410 1 (𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  {cab 2596   ⊆ wss 3540  ∩ cint 4405 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-in 3547  df-ss 3554  df-int 4406 This theorem is referenced by:  trcleq1  13525
 Copyright terms: Public domain W3C validator