Theorem uniqs2 6842

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

Step	Hyp	Ref	Expression
1		qsss.2	. . . . 5 |- ( ph -> R e. V )
2		uniqs 6840	. . . . 5 |- ( R e. V -> U. ( A /. R ) = ( R " A ) )
3	1, 2	syl 14	. . . 4 |- ( ph -> U. ( A /. R ) = ( R " A ) )
4		qsss.1	. . . . . 6 |- ( ph -> R Er A )
5		erdm 6790	. . . . . 6 |- ( R Er A -> dom R = A )
6	4, 5	syl 14	. . . . 5 |- ( ph -> dom R = A )
7	6	imaeq2d 5106	. . . 4 |- ( ph -> ( R " dom R ) = ( R " A ) )
8	3, 7	eqtr4d 2270	. . 3 |- ( ph -> U. ( A /. R ) = ( R " dom R ) )
9		imadmrn 5116	. . 3 |- ( R " dom R ) = ran R
10	8, 9	eqtrdi 2283	. 2 |- ( ph -> U. ( A /. R ) = ran R )
11		errn 6802	. . 3 |- ( R Er A -> ran R = A )
12	4, 11	syl 14	. 2 |- ( ph -> ran R = A )
13	10, 12	eqtrd 2267	1 |- ( ph -> U. ( A /. R ) = A )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   U.cuni 3919   dom cdm 4754   ran crn 4755   "cima 4757   Er wer 6777   /.cqs 6779

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-er 6780  df-ec 6782  df-qs 6786

This theorem is referenced by: (None)