Theorem qsss 6593

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 2740	. . . 4 |- x e. _V
2	1	elqs 6585	. . 3 |- ( x e. ( A /. R ) <-> E. y e. A x = [ y ] R )
3		qsss.1	. . . . . . 7 |- ( ph -> R Er A )
4	3	ecss 6575	. . . . . 6 |- ( ph -> [ y ] R C_ A )
5		sseq1 3178	. . . . . 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 3582	. . . . 5 |- ( x e. ~P A <-> x C_ A )
8	6, 7	syl6ibr 162	. . . 4 |- ( ph -> ( x = [ y ] R -> x e. ~P A ) )
9	8	rexlimdvw 2598	. . 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 3161	1 |- ( ph -> ( A /. R ) C_ ~P A )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   E.wrex 2456   C_ wss 3129   ~Pcpw 3575   Er wer 6531   [cec 6532   /.cqs 6533

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-er 6534  df-ec 6536  df-qs 6540

This theorem is referenced by:  axcnex  7857