Theorem uniqs 6857

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 6801	. . . . 5 |- ( R e. V -> [ x ] R e. _V )
2	1	ralrimivw 2624	. . . 4 |- ( R e. V -> A. x e. A [ x ] R e. _V )
3		dfiun2g 4039	. . . 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 2244	. 2 |- ( R e. V -> U. { y | E. x e. A y = [ x ] R } = U_ x e. A [ x ] R )
6		df-qs 6803	. . 3 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
7	6	unieqi 3940	. 2 |- U. ( A /. R ) = U. { y | E. x e. A y = [ x ] R }
8		df-ec 6799	. . . . 5 |- [ x ] R = ( R " { x } )
9	8	a1i 9	. . . 4 |- ( x e. A -> [ x ] R = ( R " { x } ) )
10	9	iuneq2i 4025	. . 3 |- U_ x e. A [ x ] R = U_ x e. A ( R " { x } )
11		imaiun 5956	. . 3 |- ( R " U_ x e. A { x } ) = U_ x e. A ( R " { x } )
12		iunid 4063	. . . 4 |- U_ x e. A { x } = A
13	12	imaeq2i 5119	. . 3 |- ( R " U_ x e. A { x } ) = ( R " A )
14	10, 11, 13	3eqtr2ri 2266	. 2 |- ( R " A ) = U_ x e. A [ x ] R
15	5, 7, 14	3eqtr4g 2296	1 |- ( R e. V -> U. ( A /. R ) = ( R " A ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   _Vcvv 2821   {csn 3705   U.cuni 3930   U_ciun 4007   "cima 4772   [cec 6795   /.cqs 6796

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-ec 6799  df-qs 6803

This theorem is referenced by:  uniqs2  6859  ecqs  6861