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Theorem ercl 8467
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 8465 . . . 4 (𝑅 Er 𝑋 → Rel 𝑅)
31, 2syl 17 . . 3 (𝜑 → Rel 𝑅)
4 ersym.2 . . 3 (𝜑𝐴𝑅𝐵)
5 releldm 5842 . . 3 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
63, 4, 5syl2anc 583 . 2 (𝜑𝐴 ∈ dom 𝑅)
7 erdm 8466 . . 3 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
81, 7syl 17 . 2 (𝜑 → dom 𝑅 = 𝑋)
96, 8eleqtrd 2841 1 (𝜑𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2108   class class class wbr 5070  dom cdm 5580  Rel wrel 5585   Er wer 8453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-dm 5590  df-er 8456
This theorem is referenced by:  ercl2  8469  erthi  8507  qliftfun  8549  efgcpbl2  19278  frgpcpbl  19280  prter3  36823
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