Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqvrelthi Structured version   Visualization version   GIF version

Theorem eqvrelthi 37104
Description: Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
Hypotheses
Ref Expression
eqvrelthi.1 (𝜑 → EqvRel 𝑅)
eqvrelthi.2 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
eqvrelthi (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Proof of Theorem eqvrelthi
StepHypRef Expression
1 eqvrelthi.2 . 2 (𝜑𝐴𝑅𝐵)
2 eqvrelthi.1 . . 3 (𝜑 → EqvRel 𝑅)
32, 1eqvrelcl 37103 . . 3 (𝜑𝐴 ∈ dom 𝑅)
42, 3eqvrelth 37102 . 2 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
51, 4mpbid 231 1 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542   class class class wbr 5110  [cec 8653   EqvRel weqvrel 36680
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8657  df-refrel 37003  df-symrel 37035  df-trrel 37065  df-eqvrel 37076
This theorem is referenced by:  eqvreldisj  37105  eqvrelqsel  37107
  Copyright terms: Public domain W3C validator