Theorem supfil 20902

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 3487	. . . . 5 |- ( x = y -> ( B C_ x <-> B C_ y ) )
2	1	elrab 3230	. . . 4 |- ( y e. { x e. ~P A | B C_ x } <-> ( y e. ~P A /\ B C_ y ) )
3		selpw 3987	. . . . 5 |- ( y e. ~P A <-> y C_ A )
4	3	anbi1i 700	. . . 4 |- ( ( y e. ~P A /\ B C_ y ) <-> ( y C_ A /\ B C_ y ) )
5	2, 4	bitri 253	. . 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 3091	. . 3 |- ( A e. V -> A e. _V )
8	7	3ad2ant1 1027	. 2 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> A e. _V )
9		simp2 1007	. . 3 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> B C_ A )
10		sseq2 3487	. . . . 5 |- ( y = A -> ( B C_ y <-> B C_ A ) )
11	10	sbcieg 3333	. . . 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 236	. 2 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> [. A / y ]. B C_ y )
14		ss0 3794	. . . . 5 |- ( B C_ (/) -> B = (/) )
15	14	necon3ai 2653	. . . 4 |- ( B =/= (/) -> -. B C_ (/) )
16	15	3ad2ant3 1029	. . 3 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> -. B C_ (/) )
17		0ex 4554	. . . 4 |- (/) e. _V
18		sseq2 3487	. . . 4 |- ( y = (/) -> ( B C_ y <-> B C_ (/) ) )
19	17, 18	sbcie 3335	. . 3 |- ( [. (/) / y ]. B C_ y <-> B C_ (/) )
20	16, 19	sylnibr 307	. 2 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> -. [. (/) / y ]. B C_ y )
21		sstr 3473	. . . . 5 |- ( ( B C_ w /\ w C_ z ) -> B C_ z )
22	21	expcom 437	. . . 4 |- ( w C_ z -> ( B C_ w -> B C_ z ) )
23		vex 3085	. . . . 5 |- w e. _V
24		sseq2 3487	. . . . 5 |- ( y = w -> ( B C_ y <-> B C_ w ) )
25	23, 24	sbcie 3335	. . . 4 |- ( [. w / y ]. B C_ y <-> B C_ w )
26		vex 3085	. . . . 5 |- z e. _V
27		sseq2 3487	. . . . 5 |- ( y = z -> ( B C_ y <-> B C_ z ) )
28	26, 27	sbcie 3335	. . . 4 |- ( [. z / y ]. B C_ y <-> B C_ z )
29	22, 25, 28	3imtr4g 274	. . 3 |- ( w C_ z -> ( [. w / y ]. B C_ y -> [. z / y ]. B C_ y ) )
30	29	3ad2ant3 1029	. 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 3685	. . . . . 6 |- ( ( B C_ z /\ B C_ w ) <-> B C_ ( z i^i w ) )
32	31	biimpi 198	. . . . 5 |- ( ( B C_ z /\ B C_ w ) -> B C_ ( z i^i w ) )
33	28, 25, 32	syl2anb 482	. . . 4 |- ( ( [. z / y ]. B C_ y /\ [. w / y ]. B C_ y ) -> B C_ ( z i^i w ) )
34	26	inex1 4563	. . . . 5 |- ( z i^i w ) e. _V
35		sseq2 3487	. . . . 5 |- ( y = ( z i^i w ) -> ( B C_ y <-> B C_ ( z i^i w ) ) )
36	34, 35	sbcie 3335	. . . 4 |- ( [. ( z i^i w ) / y ]. B C_ y <-> B C_ ( z i^i w ) )
37	33, 36	sylibr 216	. . 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 20865	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 188   /\ wa 371   /\ w3a 983   e. wcel 1869   =/= wne 2619   {crab 2780   _Vcvv 3082   [.wsbc 3300   i^i cin 3436   C_ wss 3437   (/)c0 3762   ~Pcpw 3980   ` cfv 5599   Filcfil 20852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fv 5607  df-fbas 18960  df-fil 20853

This theorem is referenced by:  fclscf  21032  flimfnfcls  21035