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Theorem ercl 6053
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 (φA 𝑋)

Proof of Theorem ercl
StepHypRef Expression
1 ersym.1 . . . 4 (φ𝑅 Er 𝑋)
2 errel 6051 . . . 4 (𝑅 Er 𝑋 → Rel 𝑅)
31, 2syl 14 . . 3 (φ → Rel 𝑅)
4 ersym.2 . . 3 (φA𝑅B)
5 releldm 4512 . . 3 ((Rel 𝑅 A𝑅B) → A dom 𝑅)
63, 4, 5syl2anc 391 . 2 (φA dom 𝑅)
7 erdm 6052 . . 3 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
81, 7syl 14 . 2 (φ → dom 𝑅 = 𝑋)
96, 8eleqtrd 2113 1 (φA 𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390   class class class wbr 3755  dom cdm 4288  Rel wrel 4293   Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298  df-er 6042
This theorem is referenced by:  ercl2  6055  erthi  6088  qliftfun  6124
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