Theorem supfil 20903

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 3453	. . . . 5 |- ( x = y -> ( B C_ x <-> B C_ y ) )
2	1	elrab 3195	. . . 4 |- ( y e. { x e. ~P A | B C_ x } <-> ( y e. ~P A /\ B C_ y ) )
3		selpw 3957	. . . . 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 3053	. . 3 |- ( A e. V -> A e. _V )
8	7	3ad2ant1 1028	. 2 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> A e. _V )
9		simp2 1008	. . 3 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> B C_ A )
10		sseq2 3453	. . . . 5 |- ( y = A -> ( B C_ y <-> B C_ A ) )
11	10	sbcieg 3299	. . . 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 3764	. . . . 5 |- ( B C_ (/) -> B = (/) )
15	14	necon3ai 2648	. . . 4 |- ( B =/= (/) -> -. B C_ (/) )
16	15	3ad2ant3 1030	. . 3 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> -. B C_ (/) )
17		0ex 4534	. . . 4 |- (/) e. _V
18		sseq2 3453	. . . 4 |- ( y = (/) -> ( B C_ y <-> B C_ (/) ) )
19	17, 18	sbcie 3301	. . 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 3439	. . . . 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 3047	. . . . 5 |- w e. _V
24		sseq2 3453	. . . . 5 |- ( y = w -> ( B C_ y <-> B C_ w ) )
25	23, 24	sbcie 3301	. . . 4 |- ( [. w / y ]. B C_ y <-> B C_ w )
26		vex 3047	. . . . 5 |- z e. _V
27		sseq2 3453	. . . . 5 |- ( y = z -> ( B C_ y <-> B C_ z ) )
28	26, 27	sbcie 3301	. . . 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 1030	. 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 3653	. . . . . 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 4543	. . . . 5 |- ( z i^i w ) e. _V
35		sseq2 3453	. . . . 5 |- ( y = ( z i^i w ) -> ( B C_ y <-> B C_ ( z i^i w ) ) )
36	34, 35	sbcie 3301	. . . 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 20866	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 984   e. wcel 1886   =/= wne 2621   {crab 2740   _Vcvv 3044   [.wsbc 3266   i^i cin 3402   C_ wss 3403   (/)c0 3730   ~Pcpw 3950   ` cfv 5581   Filcfil 20853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fv 5589  df-fbas 18960  df-fil 20854

This theorem is referenced by:  fclscf  21033  flimfnfcls  21036