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Theorem uniqs2 6594
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 6592 . . . . 5  |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A
) )
31, 2syl 14 . . . 4  |-  ( ph  ->  U. ( A /. R )  =  ( R " A ) )
4 qsss.1 . . . . . 6  |-  ( ph  ->  R  Er  A )
5 erdm 6544 . . . . . 6  |-  ( R  Er  A  ->  dom  R  =  A )
64, 5syl 14 . . . . 5  |-  ( ph  ->  dom  R  =  A )
76imaeq2d 4970 . . . 4  |-  ( ph  ->  ( R " dom  R )  =  ( R
" A ) )
83, 7eqtr4d 2213 . . 3  |-  ( ph  ->  U. ( A /. R )  =  ( R " dom  R
) )
9 imadmrn 4980 . . 3  |-  ( R
" dom  R )  =  ran  R
108, 9eqtrdi 2226 . 2  |-  ( ph  ->  U. ( A /. R )  =  ran  R )
11 errn 6556 . . 3  |-  ( R  Er  A  ->  ran  R  =  A )
124, 11syl 14 . 2  |-  ( ph  ->  ran  R  =  A )
1310, 12eqtrd 2210 1  |-  ( ph  ->  U. ( A /. R )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   U.cuni 3809   dom cdm 4626   ran crn 4627   "cima 4629    Er wer 6531   /.cqs 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-er 6534  df-ec 6536  df-qs 6540
This theorem is referenced by: (None)
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