Theorem qsss 6741

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 2802	. . . 4 |- x e. _V
2	1	elqs 6733	. . 3 |- ( x e. ( A /. R ) <-> E. y e. A x = [ y ] R )
3		qsss.1	. . . . . . 7 |- ( ph -> R Er A )
4	3	ecss 6723	. . . . . 6 |- ( ph -> [ y ] R C_ A )
5		sseq1 3247	. . . . . 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 3656	. . . . 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 2652	. . 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 3230	1 |- ( ph -> ( A /. R ) C_ ~P A )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   ~Pcpw 3649   Er wer 6677   [cec 6678   /.cqs 6679

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-er 6680  df-ec 6682  df-qs 6686

This theorem is referenced by:  axcnex  8046