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Theorem ecss 7955
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 7913 . . 3 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 imassrn 5635 . . 3 (𝑅 “ {𝐴}) ⊆ ran 𝑅
31, 2eqsstri 3776 . 2 [𝐴]𝑅 ⊆ ran 𝑅
4 ecss.1 . . 3 (𝜑𝑅 Er 𝑋)
5 errn 7933 . . 3 (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
64, 5syl 17 . 2 (𝜑 → ran 𝑅 = 𝑋)
73, 6syl5sseq 3794 1 (𝜑 → [𝐴]𝑅𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wss 3715  {csn 4321  ran crn 5267  cima 5269   Er wer 7908  [cec 7909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-er 7911  df-ec 7913
This theorem is referenced by:  qsss  7975  divsfval  16409  sylow1lem5  18217  sylow2alem2  18233  sylow2blem1  18235  sylow3lem3  18244  vitalilem2  23577
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