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Theorem ercl 8021
 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 8019 . . . 4 (𝑅 Er 𝑋 → Rel 𝑅)
31, 2syl 17 . . 3 (𝜑 → Rel 𝑅)
4 ersym.2 . . 3 (𝜑𝐴𝑅𝐵)
5 releldm 5592 . . 3 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
63, 4, 5syl2anc 581 . 2 (𝜑𝐴 ∈ dom 𝑅)
7 erdm 8020 . . 3 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
81, 7syl 17 . 2 (𝜑 → dom 𝑅 = 𝑋)
96, 8eleqtrd 2909 1 (𝜑𝐴𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166   class class class wbr 4874  dom cdm 5343  Rel wrel 5348   Er wer 8007 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-xp 5349  df-rel 5350  df-dm 5353  df-er 8010 This theorem is referenced by:  ercl2  8023  erthi  8059  qliftfun  8098  efgcpbl2  18524  frgpcpbl  18526  prter3  34958
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