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Theorem ecss 6938
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 6899 . . 3  |-  [ A ] R  =  ( R " { A }
)
2 imassrn 5208 . . 3  |-  ( R
" { A }
)  C_  ran  R
31, 2eqsstri 3370 . 2  |-  [ A ] R  C_  ran  R
4 ecss.1 . . 3  |-  ( ph  ->  R  Er  X )
5 errn 6919 . . 3  |-  ( R  Er  X  ->  ran  R  =  X )
64, 5syl 16 . 2  |-  ( ph  ->  ran  R  =  X )
73, 6syl5sseq 3388 1  |-  ( ph  ->  [ A ] R  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    C_ wss 3312   {csn 3806   ran crn 4871   "cima 4873    Er wer 6894   [cec 6895
This theorem is referenced by:  qsss  6957  divsfval  13764  sylow1lem5  15228  sylow2alem2  15244  sylow2blem1  15246  sylow3lem3  15255  vitalilem2  19493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-er 6897  df-ec 6899
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