Theorem uniqs2 6482

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 6480	. . . . 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 6432	. . . . . 6 |- ( R Er A -> dom R = A )
6	4, 5	syl 14	. . . . 5 |- ( ph -> dom R = A )
7	6	imaeq2d 4876	. . . 4 |- ( ph -> ( R " dom R ) = ( R " A ) )
8	3, 7	eqtr4d 2173	. . 3 |- ( ph -> U. ( A /. R ) = ( R " dom R ) )
9		imadmrn 4886	. . 3 |- ( R " dom R ) = ran R
10	8, 9	syl6eq 2186	. 2 |- ( ph -> U. ( A /. R ) = ran R )
11		errn 6444	. . 3 |- ( R Er A -> ran R = A )
12	4, 11	syl 14	. 2 |- ( ph -> ran R = A )
13	10, 12	eqtrd 2170	1 |- ( ph -> U. ( A /. R ) = A )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   U.cuni 3731   dom cdm 4534   ran crn 4535   "cima 4537   Er wer 6419   /.cqs 6421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-er 6422  df-ec 6424  df-qs 6428

This theorem is referenced by: (None)