MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniqs2 Unicode version

Theorem uniqs2 6721
Description: The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
Hypotheses
Ref Expression
qsss.1  |-  ( ph  ->  R  Er  A )
qsss.2  |-  ( ph  ->  R  e.  V )
Assertion
Ref Expression
uniqs2  |-  ( ph  ->  U. ( A /. R )  =  A )

Proof of Theorem uniqs2
StepHypRef Expression
1 qsss.2 . . . . 5  |-  ( ph  ->  R  e.  V )
2 uniqs 6719 . . . . 5  |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A
) )
31, 2syl 15 . . . 4  |-  ( ph  ->  U. ( A /. R )  =  ( R " A ) )
4 qsss.1 . . . . . 6  |-  ( ph  ->  R  Er  A )
5 erdm 6670 . . . . . 6  |-  ( R  Er  A  ->  dom  R  =  A )
64, 5syl 15 . . . . 5  |-  ( ph  ->  dom  R  =  A )
76imaeq2d 5012 . . . 4  |-  ( ph  ->  ( R " dom  R )  =  ( R
" A ) )
83, 7eqtr4d 2318 . . 3  |-  ( ph  ->  U. ( A /. R )  =  ( R " dom  R
) )
9 imadmrn 5024 . . 3  |-  ( R
" dom  R )  =  ran  R
108, 9syl6eq 2331 . 2  |-  ( ph  ->  U. ( A /. R )  =  ran  R )
11 errn 6682 . . 3  |-  ( R  Er  A  ->  ran  R  =  A )
124, 11syl 15 . 2  |-  ( ph  ->  ran  R  =  A )
1310, 12eqtrd 2315 1  |-  ( ph  ->  U. ( A /. R )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   U.cuni 3827   dom cdm 4689   ran crn 4690   "cima 4692    Er wer 6657   /.cqs 6659
This theorem is referenced by:  qshash  12285  cldsubg  17793  pi1buni  18538
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-er 6660  df-ec 6662  df-qs 6666
  Copyright terms: Public domain W3C validator