Theorem uniqs2 7809

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 7807	. . . . 5 |- ( R e. V -> U. ( A /. R ) = ( R " A ) )
3	1, 2	syl 17	. . . 4 |- ( ph -> U. ( A /. R ) = ( R " A ) )
4		qsss.1	. . . . . 6 |- ( ph -> R Er A )
5		erdm 7752	. . . . . 6 |- ( R Er A -> dom R = A )
6	4, 5	syl 17	. . . . 5 |- ( ph -> dom R = A )
7	6	imaeq2d 5466	. . . 4 |- ( ph -> ( R " dom R ) = ( R " A ) )
8	3, 7	eqtr4d 2659	. . 3 |- ( ph -> U. ( A /. R ) = ( R " dom R ) )
9		imadmrn 5476	. . 3 |- ( R " dom R ) = ran R
10	8, 9	syl6eq 2672	. 2 |- ( ph -> U. ( A /. R ) = ran R )
11		errn 7764	. . 3 |- ( R Er A -> ran R = A )
12	4, 11	syl 17	. 2 |- ( ph -> ran R = A )
13	10, 12	eqtrd 2656	1 |- ( ph -> U. ( A /. R ) = A )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   U.cuni 4436   dom cdm 5114   ran crn 5115   "cima 5117   Er wer 7739   /.cqs 7741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-er 7742  df-ec 7744  df-qs 7748

This theorem is referenced by:  qshash  14559  cldsubg  21914  pi1buni  22840