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Theorem ecss 6717
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 6678 . . 3  |-  [ A ] R  =  ( R " { A }
)
2 imassrn 5041 . . 3  |-  ( R
" { A }
)  C_  ran  R
31, 2eqsstri 3221 . 2  |-  [ A ] R  C_  ran  R
4 ecss.1 . . 3  |-  ( ph  ->  R  Er  X )
5 errn 6698 . . 3  |-  ( R  Er  X  ->  ran  R  =  X )
64, 5syl 15 . 2  |-  ( ph  ->  ran  R  =  X )
73, 6syl5sseq 3239 1  |-  ( ph  ->  [ A ] R  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    C_ wss 3165   {csn 3653   ran crn 4706   "cima 4708    Er wer 6673   [cec 6674
This theorem is referenced by:  qsss  6736  divsfval  13465  sylow1lem5  14929  sylow2alem2  14945  sylow2blem1  14947  sylow3lem3  14956  vitalilem2  18980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-er 6676  df-ec 6678
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