MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ecss Structured version   Visualization version   GIF version

Theorem ecss 8792
Description: An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
ecss.1 (𝜑𝑅 Er 𝑋)
Assertion
Ref Expression
ecss (𝜑 → [𝐴]𝑅𝑋)

Proof of Theorem ecss
StepHypRef Expression
1 df-ec 8746 . . 3 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 imassrn 6091 . . 3 (𝑅 “ {𝐴}) ⊆ ran 𝑅
31, 2eqsstri 4030 . 2 [𝐴]𝑅 ⊆ ran 𝑅
4 ecss.1 . . 3 (𝜑𝑅 Er 𝑋)
5 errn 8766 . . 3 (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
64, 5syl 17 . 2 (𝜑 → ran 𝑅 = 𝑋)
73, 6sseqtrid 4048 1 (𝜑 → [𝐴]𝑅𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wss 3963  {csn 4631  ran crn 5690  cima 5692   Er wer 8741  [cec 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-er 8744  df-ec 8746
This theorem is referenced by:  qsss  8817  divsfval  17594  ghmqusnsglem1  19311  ghmquskerlem1  19314  sylow1lem5  19635  sylow2alem2  19651  sylow2blem1  19653  sylow3lem3  19662  vitalilem2  25658  qsalrel  42260
  Copyright terms: Public domain W3C validator