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Theorem ercl 8289
Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ersym.1 (𝜑𝑅 Er 𝑋)
ersym.2 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
ercl (𝜑𝐴𝑋)

Proof of Theorem ercl
StepHypRef Expression
1 ersym.1 . . . 4 (𝜑𝑅 Er 𝑋)
2 errel 8287 . . . 4 (𝑅 Er 𝑋 → Rel 𝑅)
31, 2syl 17 . . 3 (𝜑 → Rel 𝑅)
4 ersym.2 . . 3 (𝜑𝐴𝑅𝐵)
5 releldm 5807 . . 3 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
63, 4, 5syl2anc 584 . 2 (𝜑𝐴 ∈ dom 𝑅)
7 erdm 8288 . . 3 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
81, 7syl 17 . 2 (𝜑 → dom 𝑅 = 𝑋)
96, 8eleqtrd 2912 1 (𝜑𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105   class class class wbr 5057  dom cdm 5548  Rel wrel 5553   Er wer 8275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-dm 5558  df-er 8278
This theorem is referenced by:  ercl2  8291  erthi  8329  qliftfun  8371  efgcpbl2  18812  frgpcpbl  18814  prter3  35898
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