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Theorem uniqs2 6737
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 6735 . . . . 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 6686 . . . . . 6  |-  ( R  Er  A  ->  dom  R  =  A )
64, 5syl 15 . . . . 5  |-  ( ph  ->  dom  R  =  A )
76imaeq2d 5028 . . . 4  |-  ( ph  ->  ( R " dom  R )  =  ( R
" A ) )
83, 7eqtr4d 2331 . . 3  |-  ( ph  ->  U. ( A /. R )  =  ( R " dom  R
) )
9 imadmrn 5040 . . 3  |-  ( R
" dom  R )  =  ran  R
108, 9syl6eq 2344 . 2  |-  ( ph  ->  U. ( A /. R )  =  ran  R )
11 errn 6698 . . 3  |-  ( R  Er  A  ->  ran  R  =  A )
124, 11syl 15 . 2  |-  ( ph  ->  ran  R  =  A )
1310, 12eqtrd 2328 1  |-  ( ph  ->  U. ( A /. R )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   U.cuni 3843   dom cdm 4705   ran crn 4706   "cima 4708    Er wer 6673   /.cqs 6675
This theorem is referenced by:  qshash  12301  cldsubg  17809  pi1buni  18554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-er 6676  df-ec 6678  df-qs 6682
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