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Theorem ecss 6701
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  |-  ( ph  ->  R  Er  X )
Assertion
Ref Expression
ecss  |-  ( ph  ->  [ A ] R  C_  X )

Proof of Theorem ecss
StepHypRef Expression
1 df-ec 6662 . . 3  |-  [ A ] R  =  ( R " { A }
)
2 imassrn 5025 . . 3  |-  ( R
" { A }
)  C_  ran  R
31, 2eqsstri 3208 . 2  |-  [ A ] R  C_  ran  R
4 ecss.1 . . 3  |-  ( ph  ->  R  Er  X )
5 errn 6682 . . 3  |-  ( R  Er  X  ->  ran  R  =  X )
64, 5syl 15 . 2  |-  ( ph  ->  ran  R  =  X )
73, 6syl5sseq 3226 1  |-  ( ph  ->  [ A ] R  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    C_ wss 3152   {csn 3640   ran crn 4690   "cima 4692    Er wer 6657   [cec 6658
This theorem is referenced by:  qsss  6720  divsfval  13449  sylow1lem5  14913  sylow2alem2  14929  sylow2blem1  14931  sylow3lem3  14940  vitalilem2  18964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-er 6660  df-ec 6662
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