Theorem uniqs2 6368

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 6366	. . . . 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 6318	. . . . . 6 |- ( R Er A -> dom R = A )
6	4, 5	syl 14	. . . . 5 |- ( ph -> dom R = A )
7	6	imaeq2d 4789	. . . 4 |- ( ph -> ( R " dom R ) = ( R " A ) )
8	3, 7	eqtr4d 2124	. . 3 |- ( ph -> U. ( A /. R ) = ( R " dom R ) )
9		imadmrn 4799	. . 3 |- ( R " dom R ) = ran R
10	8, 9	syl6eq 2137	. 2 |- ( ph -> U. ( A /. R ) = ran R )
11		errn 6330	. . 3 |- ( R Er A -> ran R = A )
12	4, 11	syl 14	. 2 |- ( ph -> ran R = A )
13	10, 12	eqtrd 2121	1 |- ( ph -> U. ( A /. R ) = A )

Syntax hints:   -> wi 4   = wceq 1290   e. wcel 1439   U.cuni 3661   dom cdm 4454   ran crn 4455   "cima 4457   Er wer 6305   /.cqs 6307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-iun 3740  df-br 3854  df-opab 3908  df-xp 4460  df-rel 4461  df-cnv 4462  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-er 6308  df-ec 6310  df-qs 6314

This theorem is referenced by: (None)