Theorem qsss 6806

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 2806	. . . 4 |- x e. _V
2	1	elqs 6798	. . 3 |- ( x e. ( A /. R ) <-> E. y e. A x = [ y ] R )
3		qsss.1	. . . . . . 7 |- ( ph -> R Er A )
4	3	ecss 6788	. . . . . 6 |- ( ph -> [ y ] R C_ A )
5		sseq1 3251	. . . . . 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 3663	. . . . 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 2655	. . 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 3234	1 |- ( ph -> ( A /. R ) C_ ~P A )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   E.wrex 2512   C_ wss 3201   ~Pcpw 3656   Er wer 6742   [cec 6743   /.cqs 6744

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-er 6745  df-ec 6747  df-qs 6751

This theorem is referenced by:  axcnex  8122