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Theorem ecss 8331
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 8287 . . 3 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 imassrn 5927 . . 3 (𝑅 “ {𝐴}) ⊆ ran 𝑅
31, 2eqsstri 3987 . 2 [𝐴]𝑅 ⊆ ran 𝑅
4 ecss.1 . . 3 (𝜑𝑅 Er 𝑋)
5 errn 8307 . . 3 (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
64, 5syl 17 . 2 (𝜑 → ran 𝑅 = 𝑋)
73, 6sseqtrid 4005 1 (𝜑 → [𝐴]𝑅𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wss 3919  {csn 4550  ran crn 5543  cima 5545   Er wer 8282  [cec 8283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-er 8285  df-ec 8287
This theorem is referenced by:  qsss  8354  divsfval  16820  sylow1lem5  18727  sylow2alem2  18743  sylow2blem1  18745  sylow3lem3  18754  vitalilem2  24216  qsalrel  39353
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