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Theorem ecss 8695
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 8651 . . 3 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 imassrn 6025 . . 3 (𝑅 “ {𝐴}) ⊆ ran 𝑅
31, 2eqsstri 3979 . 2 [𝐴]𝑅 ⊆ ran 𝑅
4 ecss.1 . . 3 (𝜑𝑅 Er 𝑋)
5 errn 8671 . . 3 (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
64, 5syl 17 . 2 (𝜑 → ran 𝑅 = 𝑋)
73, 6sseqtrid 3997 1 (𝜑 → [𝐴]𝑅𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wss 3911  {csn 4587  ran crn 5635  cima 5637   Er wer 8646  [cec 8647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-er 8649  df-ec 8651
This theorem is referenced by:  qsss  8718  divsfval  17430  sylow1lem5  19385  sylow2alem2  19401  sylow2blem1  19403  sylow3lem3  19412  vitalilem2  24976  ghmquskerlem1  32198  qsalrel  40667
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