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Theorem uniqs2 7160
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 7158 . . . . 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 7109 . . . . . 6  |-  ( R  Er  A  ->  dom  R  =  A )
64, 5syl 16 . . . . 5  |-  ( ph  ->  dom  R  =  A )
76imaeq2d 5167 . . . 4  |-  ( ph  ->  ( R " dom  R )  =  ( R
" A ) )
83, 7eqtr4d 2476 . . 3  |-  ( ph  ->  U. ( A /. R )  =  ( R " dom  R
) )
9 imadmrn 5177 . . 3  |-  ( R
" dom  R )  =  ran  R
108, 9syl6eq 2489 . 2  |-  ( ph  ->  U. ( A /. R )  =  ran  R )
11 errn 7121 . . 3  |-  ( R  Er  A  ->  ran  R  =  A )
124, 11syl 16 . 2  |-  ( ph  ->  ran  R  =  A )
1310, 12eqtrd 2473 1  |-  ( ph  ->  U. ( A /. R )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   U.cuni 4089   dom cdm 4838   ran crn 4839   "cima 4841    Er wer 7096   /.cqs 7098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-xp 4844  df-rel 4845  df-cnv 4846  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-er 7099  df-ec 7101  df-qs 7105
This theorem is referenced by:  qshash  13288  cldsubg  19679  pi1buni  20610
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