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Theorem ecqs 6897
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 6836 . 2  |-  [ A ] R  =  ( R " { A }
)
2 ecqs.1 . . 3  |-  R  e. 
_V
3 uniqs 6893 . . 3  |-  ( R  e.  _V  ->  U. ( { A } /. R
)  =  ( R
" { A }
) )
42, 3ax-mp 8 . 2  |-  U. ( { A } /. R
)  =  ( R
" { A }
)
51, 4eqtr4i 2403 1  |-  [ A ] R  =  U. ( { A } /. R )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   _Vcvv 2892   {csn 3750   U.cuni 3950   "cima 4814   [cec 6832   /.cqs 6833
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-13 1719  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  ax-un 4634
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-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-xp 4817  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-ec 6836  df-qs 6840
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