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Theorem ersym 7751
Description: An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ersym.1 (𝜑𝑅 Er 𝑋)
ersym.2 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
ersym (𝜑𝐵𝑅𝐴)

Proof of Theorem ersym
StepHypRef Expression
1 ersym.2 . . 3 (𝜑𝐴𝑅𝐵)
2 ersym.1 . . . . . 6 (𝜑𝑅 Er 𝑋)
3 errel 7748 . . . . . 6 (𝑅 Er 𝑋 → Rel 𝑅)
42, 3syl 17 . . . . 5 (𝜑 → Rel 𝑅)
5 brrelex12 5153 . . . . 5 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
64, 1, 5syl2anc 693 . . . 4 (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7 brcnvg 5301 . . . . 5 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵𝑅𝐴𝐴𝑅𝐵))
87ancoms 469 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵𝑅𝐴𝐴𝑅𝐵))
96, 8syl 17 . . 3 (𝜑 → (𝐵𝑅𝐴𝐴𝑅𝐵))
101, 9mpbird 247 . 2 (𝜑𝐵𝑅𝐴)
11 df-er 7739 . . . . . 6 (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
1211simp3bi 1077 . . . . 5 (𝑅 Er 𝑋 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
132, 12syl 17 . . . 4 (𝜑 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
1413unssad 3788 . . 3 (𝜑𝑅𝑅)
1514ssbrd 4694 . 2 (𝜑 → (𝐵𝑅𝐴𝐵𝑅𝐴))
1610, 15mpd 15 1 (𝜑𝐵𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  Vcvv 3198  cun 3570  wss 3572   class class class wbr 4651  ccnv 5111  dom cdm 5112  ccom 5116  Rel wrel 5117   Er wer 7736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-cnv 5120  df-er 7739
This theorem is referenced by:  ercl2  7752  ersymb  7753  ertr2d  7756  ertr3d  7757  ertr4d  7758  erth  7788  erinxp  7818  nqereu  9748  nqerf  9749  1nqenq  9781  qusgrp2  17527  efginvrel2  18134  efgcpbllemb  18162  2idlcpbl  19228  tgptsmscls  21947
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