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

Theorem erthi 7657
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.)
Hypotheses
Ref Expression
erthi.1 (𝜑𝑅 Er 𝑋)
erthi.2 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
erthi (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Proof of Theorem erthi
StepHypRef Expression
1 erthi.2 . 2 (𝜑𝐴𝑅𝐵)
2 erthi.1 . . 3 (𝜑𝑅 Er 𝑋)
32, 1ercl 7617 . . 3 (𝜑𝐴𝑋)
42, 3erth 7655 . 2 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
51, 4mpbid 220 1 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474   class class class wbr 4577   Er wer 7603  [cec 7604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-er 7606  df-ec 7608
This theorem is referenced by:  erdisj  7658  qsel  7690  addsrmo  9750  mulsrmo  9751  qusgrp2  17302  frgpinv  17946  qustgpopn  21675  blpnfctr  21992  pi1inv  22591  pi1xfrf  22592  pi1xfr  22594  pi1xfrcnvlem  22595  pi1cof  22598  vitalilem3  23102  sconpi1  30281
  Copyright terms: Public domain W3C validator