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Theorem uniqs2 6966
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 6964 . . . . 5  |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A
) )
31, 2syl 16 . . . 4  |-  ( ph  ->  U. ( A /. R )  =  ( R " A ) )
4 qsss.1 . . . . . 6  |-  ( ph  ->  R  Er  A )
5 erdm 6915 . . . . . 6  |-  ( R  Er  A  ->  dom  R  =  A )
64, 5syl 16 . . . . 5  |-  ( ph  ->  dom  R  =  A )
76imaeq2d 5203 . . . 4  |-  ( ph  ->  ( R " dom  R )  =  ( R
" A ) )
83, 7eqtr4d 2471 . . 3  |-  ( ph  ->  U. ( A /. R )  =  ( R " dom  R
) )
9 imadmrn 5215 . . 3  |-  ( R
" dom  R )  =  ran  R
108, 9syl6eq 2484 . 2  |-  ( ph  ->  U. ( A /. R )  =  ran  R )
11 errn 6927 . . 3  |-  ( R  Er  A  ->  ran  R  =  A )
124, 11syl 16 . 2  |-  ( ph  ->  ran  R  =  A )
1310, 12eqtrd 2468 1  |-  ( ph  ->  U. ( A /. R )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   U.cuni 4015   dom cdm 4878   ran crn 4879   "cima 4881    Er wer 6902   /.cqs 6904
This theorem is referenced by:  qshash  12606  cldsubg  18140  pi1buni  19065
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-er 6905  df-ec 6907  df-qs 6911
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