Theorem csdfil 21698

Description: The set of all elements whose complement is dominated by the base set is a filter. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
csdfil	|- ( ( X e. dom card /\ om ~<_ X ) -> { x e. ~P X | ( X \ x ) ~< X } e. ( Fil ` X ) )

Distinct variable group:   x, X

Proof of Theorem csdfil

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difeq2 3722	. . . . . 6 |- ( x = y -> ( X \ x ) = ( X \ y ) )
2	1	breq1d 4663	. . . . 5 |- ( x = y -> ( ( X \ x ) ~< X <-> ( X \ y ) ~< X ) )
3	2	elrab 3363	. . . 4 |- ( y e. { x e. ~P X | ( X \ x ) ~< X } <-> ( y e. ~P X /\ ( X \ y ) ~< X ) )
4		selpw 4165	. . . . 5 |- ( y e. ~P X <-> y C_ X )
5	4	anbi1i 731	. . . 4 |- ( ( y e. ~P X /\ ( X \ y ) ~< X ) <-> ( y C_ X /\ ( X \ y ) ~< X ) )
6	3, 5	bitri 264	. . 3 |- ( y e. { x e. ~P X | ( X \ x ) ~< X } <-> ( y C_ X /\ ( X \ y ) ~< X ) )
7	6	a1i 11	. 2 |- ( ( X e. dom card /\ om ~<_ X ) -> ( y e. { x e. ~P X | ( X \ x ) ~< X } <-> ( y C_ X /\ ( X \ y ) ~< X ) ) )
8		elex 3212	. . 3 |- ( X e. dom card -> X e. _V )
9	8	adantr 481	. 2 |- ( ( X e. dom card /\ om ~<_ X ) -> X e. _V )
10		difid 3948	. . . 4 |- ( X \ X ) = (/)
11		infn0 8222	. . . . . 6 |- ( om ~<_ X -> X =/= (/) )
12	11	adantl 482	. . . . 5 |- ( ( X e. dom card /\ om ~<_ X ) -> X =/= (/) )
13		0sdomg 8089	. . . . . 6 |- ( X e. dom card -> ( (/) ~< X <-> X =/= (/) ) )
14	13	adantr 481	. . . . 5 |- ( ( X e. dom card /\ om ~<_ X ) -> ( (/) ~< X <-> X =/= (/) ) )
15	12, 14	mpbird 247	. . . 4 |- ( ( X e. dom card /\ om ~<_ X ) -> (/) ~< X )
16	10, 15	syl5eqbr 4688	. . 3 |- ( ( X e. dom card /\ om ~<_ X ) -> ( X \ X ) ~< X )
17		difeq2 3722	. . . . . 6 |- ( y = X -> ( X \ y ) = ( X \ X ) )
18	17	breq1d 4663	. . . . 5 |- ( y = X -> ( ( X \ y ) ~< X <-> ( X \ X ) ~< X ) )
19	18	sbcieg 3468	. . . 4 |- ( X e. dom card -> ( [. X / y ]. ( X \ y ) ~< X <-> ( X \ X ) ~< X ) )
20	19	adantr 481	. . 3 |- ( ( X e. dom card /\ om ~<_ X ) -> ( [. X / y ]. ( X \ y ) ~< X <-> ( X \ X ) ~< X ) )
21	16, 20	mpbird 247	. 2 |- ( ( X e. dom card /\ om ~<_ X ) -> [. X / y ]. ( X \ y ) ~< X )
22		sdomirr 8097	. . 3 |- -. X ~< X
23		0ex 4790	. . . . 5 |- (/) e. _V
24		difeq2 3722	. . . . . . 7 |- ( y = (/) -> ( X \ y ) = ( X \ (/) ) )
25		dif0 3950	. . . . . . 7 |- ( X \ (/) ) = X
26	24, 25	syl6eq 2672	. . . . . 6 |- ( y = (/) -> ( X \ y ) = X )
27	26	breq1d 4663	. . . . 5 |- ( y = (/) -> ( ( X \ y ) ~< X <-> X ~< X ) )
28	23, 27	sbcie 3470	. . . 4 |- ( [. (/) / y ]. ( X \ y ) ~< X <-> X ~< X )
29	28	a1i 11	. . 3 |- ( ( X e. dom card /\ om ~<_ X ) -> ( [. (/) / y ]. ( X \ y ) ~< X <-> X ~< X ) )
30	22, 29	mtbiri 317	. 2 |- ( ( X e. dom card /\ om ~<_ X ) -> -. [. (/) / y ]. ( X \ y ) ~< X )
31		simp1l 1085	. . . . . 6 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ z C_ X /\ w C_ z ) -> X e. dom card )
32		difexg 4808	. . . . . 6 |- ( X e. dom card -> ( X \ w ) e. _V )
33	31, 32	syl 17	. . . . 5 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ z C_ X /\ w C_ z ) -> ( X \ w ) e. _V )
34		sscon 3744	. . . . . 6 |- ( w C_ z -> ( X \ z ) C_ ( X \ w ) )
35	34	3ad2ant3 1084	. . . . 5 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ z C_ X /\ w C_ z ) -> ( X \ z ) C_ ( X \ w ) )
36		ssdomg 8001	. . . . 5 |- ( ( X \ w ) e. _V -> ( ( X \ z ) C_ ( X \ w ) -> ( X \ z ) ~<_ ( X \ w ) ) )
37	33, 35, 36	sylc 65	. . . 4 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ z C_ X /\ w C_ z ) -> ( X \ z ) ~<_ ( X \ w ) )
38		domsdomtr 8095	. . . . 5 |- ( ( ( X \ z ) ~<_ ( X \ w ) /\ ( X \ w ) ~< X ) -> ( X \ z ) ~< X )
39	38	ex 450	. . . 4 |- ( ( X \ z ) ~<_ ( X \ w ) -> ( ( X \ w ) ~< X -> ( X \ z ) ~< X ) )
40	37, 39	syl 17	. . 3 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ z C_ X /\ w C_ z ) -> ( ( X \ w ) ~< X -> ( X \ z ) ~< X ) )
41		vex 3203	. . . 4 |- w e. _V
42		difeq2 3722	. . . . 5 |- ( y = w -> ( X \ y ) = ( X \ w ) )
43	42	breq1d 4663	. . . 4 |- ( y = w -> ( ( X \ y ) ~< X <-> ( X \ w ) ~< X ) )
44	41, 43	sbcie 3470	. . 3 |- ( [. w / y ]. ( X \ y ) ~< X <-> ( X \ w ) ~< X )
45		vex 3203	. . . 4 |- z e. _V
46		difeq2 3722	. . . . 5 |- ( y = z -> ( X \ y ) = ( X \ z ) )
47	46	breq1d 4663	. . . 4 |- ( y = z -> ( ( X \ y ) ~< X <-> ( X \ z ) ~< X ) )
48	45, 47	sbcie 3470	. . 3 |- ( [. z / y ]. ( X \ y ) ~< X <-> ( X \ z ) ~< X )
49	40, 44, 48	3imtr4g 285	. 2 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ z C_ X /\ w C_ z ) -> ( [. w / y ]. ( X \ y ) ~< X -> [. z / y ]. ( X \ y ) ~< X ) )
50		infunsdom 9036	. . . . . 6 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ ( ( X \ z ) ~< X /\ ( X \ w ) ~< X ) ) -> ( ( X \ z ) u. ( X \ w ) ) ~< X )
51	50	ex 450	. . . . 5 |- ( ( X e. dom card /\ om ~<_ X ) -> ( ( ( X \ z ) ~< X /\ ( X \ w ) ~< X ) -> ( ( X \ z ) u. ( X \ w ) ) ~< X ) )
52		difindi 3881	. . . . . 6 |- ( X \ ( z i^i w ) ) = ( ( X \ z ) u. ( X \ w ) )
53	52	breq1i 4660	. . . . 5 |- ( ( X \ ( z i^i w ) ) ~< X <-> ( ( X \ z ) u. ( X \ w ) ) ~< X )
54	51, 53	syl6ibr 242	. . . 4 |- ( ( X e. dom card /\ om ~<_ X ) -> ( ( ( X \ z ) ~< X /\ ( X \ w ) ~< X ) -> ( X \ ( z i^i w ) ) ~< X ) )
55	54	3ad2ant1 1082	. . 3 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ z C_ X /\ w C_ X ) -> ( ( ( X \ z ) ~< X /\ ( X \ w ) ~< X ) -> ( X \ ( z i^i w ) ) ~< X ) )
56	48, 44	anbi12i 733	. . 3 |- ( ( [. z / y ]. ( X \ y ) ~< X /\ [. w / y ]. ( X \ y ) ~< X ) <-> ( ( X \ z ) ~< X /\ ( X \ w ) ~< X ) )
57	45	inex1 4799	. . . 4 |- ( z i^i w ) e. _V
58		difeq2 3722	. . . . 5 |- ( y = ( z i^i w ) -> ( X \ y ) = ( X \ ( z i^i w ) ) )
59	58	breq1d 4663	. . . 4 |- ( y = ( z i^i w ) -> ( ( X \ y ) ~< X <-> ( X \ ( z i^i w ) ) ~< X ) )
60	57, 59	sbcie 3470	. . 3 |- ( [. ( z i^i w ) / y ]. ( X \ y ) ~< X <-> ( X \ ( z i^i w ) ) ~< X )
61	55, 56, 60	3imtr4g 285	. 2 |- ( ( ( X e. dom card /\ om ~<_ X ) /\ z C_ X /\ w C_ X ) -> ( ( [. z / y ]. ( X \ y ) ~< X /\ [. w / y ]. ( X \ y ) ~< X ) -> [. ( z i^i w ) / y ]. ( X \ y ) ~< X ) )
62	7, 9, 21, 30, 49, 61	isfild 21662	1 |- ( ( X e. dom card /\ om ~<_ X ) -> { x e. ~P X | ( X \ x ) ~< X } e. ( Fil ` X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   ` cfv 5888   omcom 7065   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761   Filcfil 21649

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-cda 8990  df-fbas 19743  df-fil 21650

This theorem is referenced by:  ufilen  21734