Theorem uniqs 6439

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 6385	. . . . 5 |- ( R e. V -> [ x ] R e. _V )
2	1	ralrimivw 2478	. . . 4 |- ( R e. V -> A. x e. A [ x ] R e. _V )
3		dfiun2g 3809	. . . 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 2118	. 2 |- ( R e. V -> U. { y | E. x e. A y = [ x ] R } = U_ x e. A [ x ] R )
6		df-qs 6387	. . 3 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
7	6	unieqi 3710	. 2 |- U. ( A /. R ) = U. { y | E. x e. A y = [ x ] R }
8		df-ec 6383	. . . . 5 |- [ x ] R = ( R " { x } )
9	8	a1i 9	. . . 4 |- ( x e. A -> [ x ] R = ( R " { x } ) )
10	9	iuneq2i 3795	. . 3 |- U_ x e. A [ x ] R = U_ x e. A ( R " { x } )
11		imaiun 5613	. . 3 |- ( R " U_ x e. A { x } ) = U_ x e. A ( R " { x } )
12		iunid 3832	. . . 4 |- U_ x e. A { x } = A
13	12	imaeq2i 4835	. . 3 |- ( R " U_ x e. A { x } ) = ( R " A )
14	10, 11, 13	3eqtr2ri 2140	. 2 |- ( R " A ) = U_ x e. A [ x ] R
15	5, 7, 14	3eqtr4g 2170	1 |- ( R e. V -> U. ( A /. R ) = ( R " A ) )

Syntax hints:   -> wi 4   = wceq 1312   e. wcel 1461   {cab 2099   A.wral 2388   E.wrex 2389   _Vcvv 2655   {csn 3491   U.cuni 3700   U_ciun 3777   "cima 4500   [cec 6379   /.cqs 6380

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 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-xp 4503  df-cnv 4505  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-ec 6383  df-qs 6387

This theorem is referenced by:  uniqs2  6441  ecqs  6443