Theorem uniqs 6680

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 6624	. . . . 5 |- ( R e. V -> [ x ] R e. _V )
2	1	ralrimivw 2580	. . . 4 |- ( R e. V -> A. x e. A [ x ] R e. _V )
3		dfiun2g 3959	. . . 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 2211	. 2 |- ( R e. V -> U. { y | E. x e. A y = [ x ] R } = U_ x e. A [ x ] R )
6		df-qs 6626	. . 3 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
7	6	unieqi 3860	. 2 |- U. ( A /. R ) = U. { y | E. x e. A y = [ x ] R }
8		df-ec 6622	. . . . 5 |- [ x ] R = ( R " { x } )
9	8	a1i 9	. . . 4 |- ( x e. A -> [ x ] R = ( R " { x } ) )
10	9	iuneq2i 3945	. . 3 |- U_ x e. A [ x ] R = U_ x e. A ( R " { x } )
11		imaiun 5829	. . 3 |- ( R " U_ x e. A { x } ) = U_ x e. A ( R " { x } )
12		iunid 3983	. . . 4 |- U_ x e. A { x } = A
13	12	imaeq2i 5020	. . 3 |- ( R " U_ x e. A { x } ) = ( R " A )
14	10, 11, 13	3eqtr2ri 2233	. 2 |- ( R " A ) = U_ x e. A [ x ] R
15	5, 7, 14	3eqtr4g 2263	1 |- ( R e. V -> U. ( A /. R ) = ( R " A ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   {cab 2191   A.wral 2484   E.wrex 2485   _Vcvv 2772   {csn 3633   U.cuni 3850   U_ciun 3927   "cima 4678   [cec 6618   /.cqs 6619

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-ec 6622  df-qs 6626

This theorem is referenced by:  uniqs2  6682  ecqs  6684