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Theorem ercl 7705
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 7703 . . . 4 (𝑅 Er 𝑋 → Rel 𝑅)
31, 2syl 17 . . 3 (𝜑 → Rel 𝑅)
4 ersym.2 . . 3 (𝜑𝐴𝑅𝐵)
5 releldm 5323 . . 3 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
63, 4, 5syl2anc 692 . 2 (𝜑𝐴 ∈ dom 𝑅)
7 erdm 7704 . . 3 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
81, 7syl 17 . 2 (𝜑 → dom 𝑅 = 𝑋)
96, 8eleqtrd 2700 1 (𝜑𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987   class class class wbr 4618  dom cdm 5079  Rel wrel 5084   Er wer 7691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-dm 5089  df-er 7694
This theorem is referenced by:  ercl2  7707  erthi  7745  qliftfun  7784  efgcpbl2  18102  frgpcpbl  18104  prter3  33682
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