Theorem supfil 21699

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 3627	. . . . 5 |- ( x = y -> ( B C_ x <-> B C_ y ) )
2	1	elrab 3363	. . . 4 |- ( y e. { x e. ~P A | B C_ x } <-> ( y e. ~P A /\ B C_ y ) )
3		selpw 4165	. . . . 5 |- ( y e. ~P A <-> y C_ A )
4	3	anbi1i 731	. . . 4 |- ( ( y e. ~P A /\ B C_ y ) <-> ( y C_ A /\ B C_ y ) )
5	2, 4	bitri 264	. . 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 3212	. . 3 |- ( A e. V -> A e. _V )
8	7	3ad2ant1 1082	. 2 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> A e. _V )
9		simp2 1062	. . 3 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> B C_ A )
10		sseq2 3627	. . . . 5 |- ( y = A -> ( B C_ y <-> B C_ A ) )
11	10	sbcieg 3468	. . . 4 |- ( A e. _V -> ( [. A / y ]. B C_ y <-> B C_ A ) )
12	8, 11	syl 17	. . 3 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> ( [. A / y ]. B C_ y <-> B C_ A ) )
13	9, 12	mpbird 247	. 2 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> [. A / y ]. B C_ y )
14		ss0 3974	. . . . 5 |- ( B C_ (/) -> B = (/) )
15	14	necon3ai 2819	. . . 4 |- ( B =/= (/) -> -. B C_ (/) )
16	15	3ad2ant3 1084	. . 3 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> -. B C_ (/) )
17		0ex 4790	. . . 4 |- (/) e. _V
18		sseq2 3627	. . . 4 |- ( y = (/) -> ( B C_ y <-> B C_ (/) ) )
19	17, 18	sbcie 3470	. . 3 |- ( [. (/) / y ]. B C_ y <-> B C_ (/) )
20	16, 19	sylnibr 319	. 2 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> -. [. (/) / y ]. B C_ y )
21		sstr 3611	. . . . 5 |- ( ( B C_ w /\ w C_ z ) -> B C_ z )
22	21	expcom 451	. . . 4 |- ( w C_ z -> ( B C_ w -> B C_ z ) )
23		vex 3203	. . . . 5 |- w e. _V
24		sseq2 3627	. . . . 5 |- ( y = w -> ( B C_ y <-> B C_ w ) )
25	23, 24	sbcie 3470	. . . 4 |- ( [. w / y ]. B C_ y <-> B C_ w )
26		vex 3203	. . . . 5 |- z e. _V
27		sseq2 3627	. . . . 5 |- ( y = z -> ( B C_ y <-> B C_ z ) )
28	26, 27	sbcie 3470	. . . 4 |- ( [. z / y ]. B C_ y <-> B C_ z )
29	22, 25, 28	3imtr4g 285	. . 3 |- ( w C_ z -> ( [. w / y ]. B C_ y -> [. z / y ]. B C_ y ) )
30	29	3ad2ant3 1084	. 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 3835	. . . . . 6 |- ( ( B C_ z /\ B C_ w ) <-> B C_ ( z i^i w ) )
32	31	biimpi 206	. . . . 5 |- ( ( B C_ z /\ B C_ w ) -> B C_ ( z i^i w ) )
33	28, 25, 32	syl2anb 496	. . . 4 |- ( ( [. z / y ]. B C_ y /\ [. w / y ]. B C_ y ) -> B C_ ( z i^i w ) )
34	26	inex1 4799	. . . . 5 |- ( z i^i w ) e. _V
35		sseq2 3627	. . . . 5 |- ( y = ( z i^i w ) -> ( B C_ y <-> B C_ ( z i^i w ) ) )
36	34, 35	sbcie 3470	. . . 4 |- ( [. ( z i^i w ) / y ]. B C_ y <-> B C_ ( z i^i w ) )
37	33, 36	sylibr 224	. . 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 21662	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 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   ` cfv 5888   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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-nel 2898  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-fv 5896  df-fbas 19743  df-fil 21650

This theorem is referenced by:  fclscf  21829  flimfnfcls  21832