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Theorem ecss 6905
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 6866 . . 3  |-  [ A ] R  =  ( R " { A }
)
2 imassrn 5175 . . 3  |-  ( R
" { A }
)  C_  ran  R
31, 2eqsstri 3338 . 2  |-  [ A ] R  C_  ran  R
4 ecss.1 . . 3  |-  ( ph  ->  R  Er  X )
5 errn 6886 . . 3  |-  ( R  Er  X  ->  ran  R  =  X )
64, 5syl 16 . 2  |-  ( ph  ->  ran  R  =  X )
73, 6syl5sseq 3356 1  |-  ( ph  ->  [ A ] R  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    C_ wss 3280   {csn 3774   ran crn 4838   "cima 4840    Er wer 6861   [cec 6862
This theorem is referenced by:  qsss  6924  divsfval  13727  sylow1lem5  15191  sylow2alem2  15207  sylow2blem1  15209  sylow3lem3  15218  vitalilem2  19454
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-er 6864  df-ec 6866
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