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Theorem ecqs 6960
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 6899 . 2  |-  [ A ] R  =  ( R " { A }
)
2 ecqs.1 . . 3  |-  R  e. 
_V
3 uniqs 6956 . . 3  |-  ( R  e.  _V  ->  U. ( { A } /. R
)  =  ( R
" { A }
) )
42, 3ax-mp 8 . 2  |-  U. ( { A } /. R
)  =  ( R
" { A }
)
51, 4eqtr4i 2458 1  |-  [ A ] R  =  U. ( { A } /. R )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806   U.cuni 4007   "cima 4873   [cec 6895   /.cqs 6896
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-13 1727  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  ax-un 4693
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-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-ec 6899  df-qs 6903
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