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Theorem ercl 7906
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 7904 . . . 4 (𝑅 Er 𝑋 → Rel 𝑅)
31, 2syl 17 . . 3 (𝜑 → Rel 𝑅)
4 ersym.2 . . 3 (𝜑𝐴𝑅𝐵)
5 releldm 5496 . . 3 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
63, 4, 5syl2anc 565 . 2 (𝜑𝐴 ∈ dom 𝑅)
7 erdm 7905 . . 3 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
81, 7syl 17 . 2 (𝜑 → dom 𝑅 = 𝑋)
96, 8eleqtrd 2851 1 (𝜑𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144   class class class wbr 4784  dom cdm 5249  Rel wrel 5254   Er wer 7892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-dm 5259  df-er 7895
This theorem is referenced by:  ercl2  7908  erthi  7944  qliftfun  7983  efgcpbl2  18376  frgpcpbl  18378  prter3  34683
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