Theorem supfil 20522

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 3521	. . . . 5 |- ( x = y -> ( B C_ x <-> B C_ y ) )
2	1	elrab 3257	. . . 4 |- ( y e. { x e. ~P A | B C_ x } <-> ( y e. ~P A /\ B C_ y ) )
3		selpw 4022	. . . . 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 3118	. . 3 |- ( A e. V -> A e. _V )
8	7	3ad2ant1 1017	. 2 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> A e. _V )
9		simp2 997	. . 3 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> B C_ A )
10		sseq2 3521	. . . . 5 |- ( y = A -> ( B C_ y <-> B C_ A ) )
11	10	sbcieg 3360	. . . 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 3825	. . . . 5 |- ( B C_ (/) -> B = (/) )
15	14	necon3ai 2685	. . . 4 |- ( B =/= (/) -> -. B C_ (/) )
16	15	3ad2ant3 1019	. . 3 |- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> -. B C_ (/) )
17		0ex 4587	. . . 4 |- (/) e. _V
18		sseq2 3521	. . . 4 |- ( y = (/) -> ( B C_ y <-> B C_ (/) ) )
19	17, 18	sbcie 3362	. . 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 3507	. . . . 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 3112	. . . . 5 |- w e. _V
24		sseq2 3521	. . . . 5 |- ( y = w -> ( B C_ y <-> B C_ w ) )
25	23, 24	sbcie 3362	. . . 4 |- ( [. w / y ]. B C_ y <-> B C_ w )
26		vex 3112	. . . . 5 |- z e. _V
27		sseq2 3521	. . . . 5 |- ( y = z -> ( B C_ y <-> B C_ z ) )
28	26, 27	sbcie 3362	. . . 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 1019	. 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 3716	. . . . . 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 4597	. . . . 5 |- ( z i^i w ) e. _V
35		sseq2 3521	. . . . 5 |- ( y = ( z i^i w ) -> ( B C_ y <-> B C_ ( z i^i w ) ) )
36	34, 35	sbcie 3362	. . . 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 20485	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 973   e. wcel 1819   =/= wne 2652   {crab 2811   _Vcvv 3109   [.wsbc 3327   i^i cin 3470   C_ wss 3471   (/)c0 3793   ~Pcpw 4015   ` cfv 5594   Filcfil 20472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-fbas 18543  df-fil 20473

This theorem is referenced by:  fclscf  20652  flimfnfcls  20655