Theorem uniqs2 6705

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 6703	. . . . 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 6653	. . . . . 6 |- ( R Er A -> dom R = A )
6	4, 5	syl 14	. . . . 5 |- ( ph -> dom R = A )
7	6	imaeq2d 5041	. . . 4 |- ( ph -> ( R " dom R ) = ( R " A ) )
8	3, 7	eqtr4d 2243	. . 3 |- ( ph -> U. ( A /. R ) = ( R " dom R ) )
9		imadmrn 5051	. . 3 |- ( R " dom R ) = ran R
10	8, 9	eqtrdi 2256	. 2 |- ( ph -> U. ( A /. R ) = ran R )
11		errn 6665	. . 3 |- ( R Er A -> ran R = A )
12	4, 11	syl 14	. 2 |- ( ph -> ran R = A )
13	10, 12	eqtrd 2240	1 |- ( ph -> U. ( A /. R ) = A )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   U.cuni 3864   dom cdm 4693   ran crn 4694   "cima 4696   Er wer 6640   /.cqs 6642

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-er 6643  df-ec 6645  df-qs 6649

This theorem is referenced by: (None)