Theorem supfil 19595

Description: The supersets of a nonempty set which are also subsets of a given base set form a filter. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)

Assertion

Ref	Expression
supfil	|- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> { x e. ~P A | B C_ x } e. ( Fil ` A ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem supfil

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

Step	Hyp	Ref	Expression
1		sseq2 3481	. . . . 5 |- ( x = y -> ( B C_ x <-> B C_ y ) )
2	1	elrab 3218	. . . 4 |- ( y e. { x e. ~P A | B C_ x } <-> ( y e. ~P A /\ B C_ y ) )
3		selpw 3970	. . . . 5 |- ( y e. ~P A <-> y C_ A )
4	3	anbi1i 695	. . . 4 |- ( ( y e. ~P A /\ B C_ y ) <-> ( y C_ A /\ B C_ y ) )
5	2, 4	bitri 249	. . 3 |- ( y e. { x e. ~P A | B C_ x } <-> ( y C_ A /\ B C_ y ) )
6	5	a1i 11	. 2 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> ( y e. { x e. ~P A | B C_ x } <-> ( y C_ A /\ B C_ y ) ) )
7		elex 3081	. . 3 |- ( A e. V -> A e. _V )
8	7	3ad2ant1 1009	. 2 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> A e. _V )
9		simp2 989	. . 3 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> B C_ A )
10		sseq2 3481	. . . . 5 |- ( y = A -> ( B C_ y <-> B C_ A ) )
11	10	sbcieg 3321	. . . 4 |- ( A e. _V -> ( [. A / y ]. B C_ y <-> B C_ A ) )
12	8, 11	syl 16	. . 3 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> ( [. A / y ]. B C_ y <-> B C_ A ) )
13	9, 12	mpbird 232	. 2 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> [. A / y ]. B C_ y )
14		ss0 3771	. . . . 5 |- ( B C_ (/) -> B = (/) )
15	14	necon3ai 2677	. . . 4 |- ( B =/= (/) -> -. B C_ (/) )
16	15	3ad2ant3 1011	. . 3 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> -. B C_ (/) )
17		0ex 4525	. . . 4 |- (/) e. _V
18		sseq2 3481	. . . 4 |- ( y = (/) -> ( B C_ y <-> B C_ (/) ) )
19	17, 18	sbcie 3323	. . 3 |- ( [. (/) / y ]. B C_ y <-> B C_ (/) )
20	16, 19	sylnibr 305	. 2 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> -. [. (/) / y ]. B C_ y )
21		sstr 3467	. . . . 5 |- ( ( B C_ w /\ w C_ z ) -> B C_ z )
22	21	expcom 435	. . . 4 |- ( w C_ z -> ( B C_ w -> B C_ z ) )
23		vex 3075	. . . . 5 |- w e. _V
24		sseq2 3481	. . . . 5 |- ( y = w -> ( B C_ y <-> B C_ w ) )
25	23, 24	sbcie 3323	. . . 4 |- ( [. w / y ]. B C_ y <-> B C_ w )
26		vex 3075	. . . . 5 |- z e. _V
27		sseq2 3481	. . . . 5 |- ( y = z -> ( B C_ y <-> B C_ z ) )
28	26, 27	sbcie 3323	. . . 4 |- ( [. z / y ]. B C_ y <-> B C_ z )
29	22, 25, 28	3imtr4g 270	. . 3 |- ( w C_ z -> ( [. w / y ]. B C_ y -> [. z / y ]. B C_ y ) )
30	29	3ad2ant3 1011	. 2 |- ( ( ( A e. V /\ B C_ A /\ B =/= (/) ) /\ z C_ A /\ w C_ z ) -> ( [. w / y ]. B C_ y -> [. z / y ]. B C_ y ) )
31		ssin 3675	. . . . . 6 |- ( ( B C_ z /\ B C_ w ) <-> B C_ ( z i^i w ) )
32	31	biimpi 194	. . . . 5 |- ( ( B C_ z /\ B C_ w ) -> B C_ ( z i^i w ) )
33	28, 25, 32	syl2anb 479	. . . 4 |- ( ( [. z / y ]. B C_ y /\ [. w / y ]. B C_ y ) -> B C_ ( z i^i w ) )
34	26	inex1 4536	. . . . 5 |- ( z i^i w ) e. _V
35		sseq2 3481	. . . . 5 |- ( y = ( z i^i w ) -> ( B C_ y <-> B C_ ( z i^i w ) ) )
36	34, 35	sbcie 3323	. . . 4 |- ( [. ( z i^i w ) / y ]. B C_ y <-> B C_ ( z i^i w ) )
37	33, 36	sylibr 212	. . 3 |- ( ( [. z / y ]. B C_ y /\ [. w / y ]. B C_ y ) -> [. ( z i^i w ) / y ]. B C_ y )
38	37	a1i 11	. 2 |- ( ( ( A e. V /\ B C_ A /\ B =/= (/) ) /\ z C_ A /\ w C_ A ) -> ( ( [. z / y ]. B C_ y /\ [. w / y ]. B C_ y ) -> [. ( z i^i w ) / y ]. B C_ y ) )
39	6, 8, 13, 20, 30, 38	isfild 19558	1 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> { x e. ~P A | B C_ x } e. ( Fil ` A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   e. wcel 1758   =/= wne 2645   {crab 2800   _Vcvv 3072   [.wsbc 3288   i^i cin 3430   C_ wss 3431   (/)c0 3740   ~Pcpw 3963   ` cfv 5521   Filcfil 19545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fv 5529  df-fbas 17934  df-fil 19546

This theorem is referenced by:  fclscf  19725  flimfnfcls  19728