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Theorem qsexg 6954
Description: A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
qsexg  |-  ( A  e.  V  ->  ( A /. R )  e. 
_V )

Proof of Theorem qsexg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-qs 6903 . 2  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
2 abrexexg 5976 . 2  |-  ( A  e.  V  ->  { y  |  E. x  e.  A  y  =  [
x ] R }  e.  _V )
31, 2syl5eqel 2519 1  |-  ( A  e.  V  ->  ( A /. R )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   _Vcvv 2948   [cec 6895   /.cqs 6896
This theorem is referenced by:  qsex  6955  pstmval  24282  pstmxmet  24284
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-rep 4312  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-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  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-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-qs 6903
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