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Theorem ecss 8133
 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 8089 . . 3 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 imassrn 5778 . . 3 (𝑅 “ {𝐴}) ⊆ ran 𝑅
31, 2eqsstri 3884 . 2 [𝐴]𝑅 ⊆ ran 𝑅
4 ecss.1 . . 3 (𝜑𝑅 Er 𝑋)
5 errn 8109 . . 3 (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
64, 5syl 17 . 2 (𝜑 → ran 𝑅 = 𝑋)
73, 6syl5sseq 3902 1 (𝜑 → [𝐴]𝑅𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1508   ⊆ wss 3822  {csn 4435  ran crn 5404   “ cima 5406   Er wer 8084  [cec 8085 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-xp 5409  df-rel 5410  df-cnv 5411  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-er 8087  df-ec 8089 This theorem is referenced by:  qsss  8156  divsfval  16674  sylow1lem5  18500  sylow2alem2  18516  sylow2blem1  18518  sylow3lem3  18527  vitalilem2  23928  qsalrel  38608
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