ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ercl GIF version

Theorem ercl 6446
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 6444 . . . 4 (𝑅 Er 𝑋 → Rel 𝑅)
31, 2syl 14 . . 3 (𝜑 → Rel 𝑅)
4 ersym.2 . . 3 (𝜑𝐴𝑅𝐵)
5 releldm 4780 . . 3 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
63, 4, 5syl2anc 409 . 2 (𝜑𝐴 ∈ dom 𝑅)
7 erdm 6445 . . 3 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
81, 7syl 14 . 2 (𝜑 → dom 𝑅 = 𝑋)
96, 8eleqtrd 2219 1 (𝜑𝐴𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  wcel 1481   class class class wbr 3935  dom cdm 4545  Rel wrel 4550   Er wer 6432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-br 3936  df-opab 3996  df-xp 4551  df-rel 4552  df-dm 4555  df-er 6435
This theorem is referenced by:  ercl2  6448  erthi  6481  qliftfun  6517
  Copyright terms: Public domain W3C validator