Theorem uniqs 6187

Description: The union of a quotient set. (Contributed by NM, 9-Dec-2008.)

Assertion

Ref	Expression
uniqs	|- ( R e. V -> U. ( A /. R ) = ( R " A ) )

Proof of Theorem uniqs

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecexg 6133	. . . . 5 |- ( R e. V -> [ x ] R e. _V )
2	1	ralrimivw 2435	. . . 4 |- ( R e. V -> A. x e. A [ x ] R e. _V )
3		dfiun2g 3710	. . . 4 |- ( A. x e. A [ x ] R e. _V -> U_ x e. A [ x ] R = U. { y | E. x e. A y = [ x ] R } )
4	2, 3	syl 14	. . 3 |- ( R e. V -> U_ x e. A [ x ] R = U. { y | E. x e. A y = [ x ] R } )
5	4	eqcomd 2086	. 2 |- ( R e. V -> U. { y | E. x e. A y = [ x ] R } = U_ x e. A [ x ] R )
6		df-qs 6135	. . 3 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
7	6	unieqi 3611	. 2 |- U. ( A /. R ) = U. { y | E. x e. A y = [ x ] R }
8		df-ec 6131	. . . . 5 |- [ x ] R = ( R " { x } )
9	8	a1i 9	. . . 4 |- ( x e. A -> [ x ] R = ( R " { x } ) )
10	9	iuneq2i 3696	. . 3 |- U_ x e. A [ x ] R = U_ x e. A ( R " { x } )
11		imaiun 5420	. . 3 |- ( R " U_ x e. A { x } ) = U_ x e. A ( R " { x } )
12		iunid 3733	. . . 4 |- U_ x e. A { x } = A
13	12	imaeq2i 4686	. . 3 |- ( R " U_ x e. A { x } ) = ( R " A )
14	10, 11, 13	3eqtr2ri 2108	. 2 |- ( R " A ) = U_ x e. A [ x ] R
15	5, 7, 14	3eqtr4g 2138	1 |- ( R e. V -> U. ( A /. R ) = ( R " A ) )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   E.wrex 2349   _Vcvv 2601   {csn 3398   U.cuni 3601   U_ciun 3678   "cima 4366   [cec 6127   /.cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-ec 6131  df-qs 6135

This theorem is referenced by:  uniqs2  6189  ecqs  6191