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Theorem ecss 6875
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 6836 . . 3  |-  [ A ] R  =  ( R " { A }
)
2 imassrn 5149 . . 3  |-  ( R
" { A }
)  C_  ran  R
31, 2eqsstri 3314 . 2  |-  [ A ] R  C_  ran  R
4 ecss.1 . . 3  |-  ( ph  ->  R  Er  X )
5 errn 6856 . . 3  |-  ( R  Er  X  ->  ran  R  =  X )
64, 5syl 16 . 2  |-  ( ph  ->  ran  R  =  X )
73, 6syl5sseq 3332 1  |-  ( ph  ->  [ A ] R  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    C_ wss 3256   {csn 3750   ran crn 4812   "cima 4814    Er wer 6831   [cec 6832
This theorem is referenced by:  qsss  6894  divsfval  13692  sylow1lem5  15156  sylow2alem2  15172  sylow2blem1  15174  sylow3lem3  15183  vitalilem2  19361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-er 6834  df-ec 6836
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