Theorem qsss 6648

Description: A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
qsss.1	|- ( ph -> R Er A )

Assertion

Ref	Expression
qsss	|- ( ph -> ( A /. R ) C_ ~P A )

Proof of Theorem qsss

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

Step	Hyp	Ref	Expression
1		vex 2763	. . . 4 |- x e. _V
2	1	elqs 6640	. . 3 |- ( x e. ( A /. R ) <-> E. y e. A x = [ y ] R )
3		qsss.1	. . . . . . 7 |- ( ph -> R Er A )
4	3	ecss 6630	. . . . . 6 |- ( ph -> [ y ] R C_ A )
5		sseq1 3202	. . . . . 6 |- ( x = [ y ] R -> ( x C_ A <-> [ y ] R C_ A ) )
6	4, 5	syl5ibrcom 157	. . . . 5 |- ( ph -> ( x = [ y ] R -> x C_ A ) )
7		velpw 3608	. . . . 5 |- ( x e. ~P A <-> x C_ A )
8	6, 7	imbitrrdi 162	. . . 4 |- ( ph -> ( x = [ y ] R -> x e. ~P A ) )
9	8	rexlimdvw 2615	. . 3 |- ( ph -> ( E. y e. A x = [ y ] R -> x e. ~P A ) )
10	2, 9	biimtrid 152	. 2 |- ( ph -> ( x e. ( A /. R ) -> x e. ~P A ) )
11	10	ssrdv 3185	1 |- ( ph -> ( A /. R ) C_ ~P A )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   E.wrex 2473   C_ wss 3153   ~Pcpw 3601   Er wer 6584   [cec 6585   /.cqs 6586

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-er 6587  df-ec 6589  df-qs 6593

This theorem is referenced by:  axcnex  7919