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Theorem ecqs 6927
Description: Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
Hypothesis
Ref Expression
ecqs.1  |-  R  e. 
_V
Assertion
Ref Expression
ecqs  |-  [ A ] R  =  U. ( { A } /. R )

Proof of Theorem ecqs
StepHypRef Expression
1 df-ec 6866 . 2  |-  [ A ] R  =  ( R " { A }
)
2 ecqs.1 . . 3  |-  R  e. 
_V
3 uniqs 6923 . . 3  |-  ( R  e.  _V  ->  U. ( { A } /. R
)  =  ( R
" { A }
) )
42, 3ax-mp 8 . 2  |-  U. ( { A } /. R
)  =  ( R
" { A }
)
51, 4eqtr4i 2427 1  |-  [ A ] R  =  U. ( { A } /. R )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2916   {csn 3774   U.cuni 3975   "cima 4840   [cec 6862   /.cqs 6863
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-13 1723  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  ax-un 4660
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-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ec 6866  df-qs 6870
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