Theorem uniqs 6703

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 6647	. . . . 5 |- ( R e. V -> [ x ] R e. _V )
2	1	ralrimivw 2582	. . . 4 |- ( R e. V -> A. x e. A [ x ] R e. _V )
3		dfiun2g 3973	. . . 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 2213	. 2 |- ( R e. V -> U. { y | E. x e. A y = [ x ] R } = U_ x e. A [ x ] R )
6		df-qs 6649	. . 3 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
7	6	unieqi 3874	. 2 |- U. ( A /. R ) = U. { y | E. x e. A y = [ x ] R }
8		df-ec 6645	. . . . 5 |- [ x ] R = ( R " { x } )
9	8	a1i 9	. . . 4 |- ( x e. A -> [ x ] R = ( R " { x } ) )
10	9	iuneq2i 3959	. . 3 |- U_ x e. A [ x ] R = U_ x e. A ( R " { x } )
11		imaiun 5852	. . 3 |- ( R " U_ x e. A { x } ) = U_ x e. A ( R " { x } )
12		iunid 3997	. . . 4 |- U_ x e. A { x } = A
13	12	imaeq2i 5039	. . 3 |- ( R " U_ x e. A { x } ) = ( R " A )
14	10, 11, 13	3eqtr2ri 2235	. 2 |- ( R " A ) = U_ x e. A [ x ] R
15	5, 7, 14	3eqtr4g 2265	1 |- ( R e. V -> U. ( A /. R ) = ( R " A ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   _Vcvv 2776   {csn 3643   U.cuni 3864   U_ciun 3941   "cima 4696   [cec 6641   /.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-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-ec 6645  df-qs 6649

This theorem is referenced by:  uniqs2  6705  ecqs  6707