Theorem qsss 6367

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 2625	. . . 4 |- x e. _V
2	1	elqs 6359	. . 3 |- ( x e. ( A /. R ) <-> E. y e. A x = [ y ] R )
3		qsss.1	. . . . . . 7 |- ( ph -> R Er A )
4	3	ecss 6349	. . . . . 6 |- ( ph -> [ y ] R C_ A )
5		sseq1 3050	. . . . . 6 |- ( x = [ y ] R -> ( x C_ A <-> [ y ] R C_ A ) )
6	4, 5	syl5ibrcom 156	. . . . 5 |- ( ph -> ( x = [ y ] R -> x C_ A ) )
7		selpw 3442	. . . . 5 |- ( x e. ~P A <-> x C_ A )
8	6, 7	syl6ibr 161	. . . 4 |- ( ph -> ( x = [ y ] R -> x e. ~P A ) )
9	8	rexlimdvw 2495	. . 3 |- ( ph -> ( E. y e. A x = [ y ] R -> x e. ~P A ) )
10	2, 9	syl5bi 151	. 2 |- ( ph -> ( x e. ( A /. R ) -> x e. ~P A ) )
11	10	ssrdv 3034	1 |- ( ph -> ( A /. R ) C_ ~P A )

Syntax hints:   -> wi 4   = wceq 1290   e. wcel 1439   E.wrex 2361   C_ wss 3002   ~Pcpw 3435   Er wer 6305   [cec 6306   /.cqs 6307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-xp 4460  df-rel 4461  df-cnv 4462  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-er 6308  df-ec 6310  df-qs 6314

This theorem is referenced by:  axcnex  7459