Theorem wdompwdom 8483

Description: Weak dominance strengthens to usual dominance on the power sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)

Assertion

Ref	Expression
wdompwdom	|- ( X ~<_* Y -> ~P X ~<_ ~P Y )

Proof of Theorem wdompwdom

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relwdom 8471	. . . . . 6 |- Rel ~<_*
2	1	brrelex2i 5159	. . . . 5 |- ( X ~<_* Y -> Y e. _V )
3		pwexg 4850	. . . . 5 |- ( Y e. _V -> ~P Y e. _V )
4	2, 3	syl 17	. . . 4 |- ( X ~<_* Y -> ~P Y e. _V )
5		0ss 3972	. . . . 5 |- (/) C_ Y
6		sspwb 4917	. . . . 5 |- ( (/) C_ Y <-> ~P (/) C_ ~P Y )
7	5, 6	mpbi 220	. . . 4 |- ~P (/) C_ ~P Y
8		ssdomg 8001	. . . 4 |- ( ~P Y e. _V -> ( ~P (/) C_ ~P Y -> ~P (/) ~<_ ~P Y ) )
9	4, 7, 8	mpisyl 21	. . 3 |- ( X ~<_* Y -> ~P (/) ~<_ ~P Y )
10		pweq 4161	. . . 4 |- ( X = (/) -> ~P X = ~P (/) )
11	10	breq1d 4663	. . 3 |- ( X = (/) -> ( ~P X ~<_ ~P Y <-> ~P (/) ~<_ ~P Y ) )
12	9, 11	syl5ibr 236	. 2 |- ( X = (/) -> ( X ~<_* Y -> ~P X ~<_ ~P Y ) )
13		brwdomn0 8474	. . 3 |- ( X =/= (/) -> ( X ~<_* Y <-> E. z z : Y -onto-> X ) )
14		vex 3203	. . . . 5 |- z e. _V
15		fopwdom 8068	. . . . 5 |- ( ( z e. _V /\ z : Y -onto-> X ) -> ~P X ~<_ ~P Y )
16	14, 15	mpan 706	. . . 4 |- ( z : Y -onto-> X -> ~P X ~<_ ~P Y )
17	16	exlimiv 1858	. . 3 |- ( E. z z : Y -onto-> X -> ~P X ~<_ ~P Y )
18	13, 17	syl6bi 243	. 2 |- ( X =/= (/) -> ( X ~<_* Y -> ~P X ~<_ ~P Y ) )
19	12, 18	pm2.61ine 2877	1 |- ( X ~<_* Y -> ~P X ~<_ ~P Y )

Syntax hints:   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   -onto->wfo 5886   ~<_ cdom 7953   ~<_^* cwdom 8462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-dom 7957  df-wdom 8464

This theorem is referenced by:  isfin32i  9187  hsmexlem1  9248  hsmexlem3  9250  gchhar  9501