Theorem supfil 23843

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 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ 𝐵 ⊆ 𝑥} ∈ (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 4003	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑦))
2	1	elrab 3679	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝐵 ⊆ 𝑥} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ 𝐵 ⊆ 𝑦))
3		velpw 4609	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
4	3	anbi1i 622	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝐵 ⊆ 𝑦) ↔ (𝑦 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝑦))
5	2, 4	bitri 274	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝐵 ⊆ 𝑥} ↔ (𝑦 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝑦))
6	5	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝐵 ⊆ 𝑥} ↔ (𝑦 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝑦)))
7		simp1 1133	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → 𝐴 ∈ 𝑉)
8		simp2 1134	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → 𝐵 ⊆ 𝐴)
9		sseq2 4003	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝐴))
10	9	sbcieg 3814	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝐴))
11	7, 10	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ([𝐴 / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝐴))
12	8, 11	mpbird 256	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → [𝐴 / 𝑦]𝐵 ⊆ 𝑦)
13		ss0 4400	. . . . 5 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅)
14	13	necon3ai 2954	. . . 4 ⊢ (𝐵 ≠ ∅ → ¬ 𝐵 ⊆ ∅)
15	14	3ad2ant3 1132	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ¬ 𝐵 ⊆ ∅)
16		0ex 5308	. . . 4 ⊢ ∅ ∈ V
17		sseq2 4003	. . . 4 ⊢ (𝑦 = ∅ → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ ∅))
18	16, 17	sbcie 3817	. . 3 ⊢ ([∅ / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ ∅)
19	15, 18	sylnibr 328	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ¬ [∅ / 𝑦]𝐵 ⊆ 𝑦)
20		sstr 3985	. . . . 5 ⊢ ((𝐵 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑧) → 𝐵 ⊆ 𝑧)
21	20	expcom 412	. . . 4 ⊢ (𝑤 ⊆ 𝑧 → (𝐵 ⊆ 𝑤 → 𝐵 ⊆ 𝑧))
22		vex 3465	. . . . 5 ⊢ 𝑤 ∈ V
23		sseq2 4003	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑤))
24	22, 23	sbcie 3817	. . . 4 ⊢ ([𝑤 / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑤)
25		vex 3465	. . . . 5 ⊢ 𝑧 ∈ V
26		sseq2 4003	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑧))
27	25, 26	sbcie 3817	. . . 4 ⊢ ([𝑧 / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑧)
28	21, 24, 27	3imtr4g 295	. . 3 ⊢ (𝑤 ⊆ 𝑧 → ([𝑤 / 𝑦]𝐵 ⊆ 𝑦 → [𝑧 / 𝑦]𝐵 ⊆ 𝑦))
29	28	3ad2ant3 1132	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑧 ⊆ 𝐴 ∧ 𝑤 ⊆ 𝑧) → ([𝑤 / 𝑦]𝐵 ⊆ 𝑦 → [𝑧 / 𝑦]𝐵 ⊆ 𝑦))
30		ssin 4229	. . . . . 6 ⊢ ((𝐵 ⊆ 𝑧 ∧ 𝐵 ⊆ 𝑤) ↔ 𝐵 ⊆ (𝑧 ∩ 𝑤))
31	30	biimpi 215	. . . . 5 ⊢ ((𝐵 ⊆ 𝑧 ∧ 𝐵 ⊆ 𝑤) → 𝐵 ⊆ (𝑧 ∩ 𝑤))
32	27, 24, 31	syl2anb 596	. . . 4 ⊢ (([𝑧 / 𝑦]𝐵 ⊆ 𝑦 ∧ [𝑤 / 𝑦]𝐵 ⊆ 𝑦) → 𝐵 ⊆ (𝑧 ∩ 𝑤))
33	25	inex1 5318	. . . . 5 ⊢ (𝑧 ∩ 𝑤) ∈ V
34		sseq2 4003	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ (𝑧 ∩ 𝑤)))
35	33, 34	sbcie 3817	. . . 4 ⊢ ([(𝑧 ∩ 𝑤) / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ (𝑧 ∩ 𝑤))
36	32, 35	sylibr 233	. . 3 ⊢ (([𝑧 / 𝑦]𝐵 ⊆ 𝑦 ∧ [𝑤 / 𝑦]𝐵 ⊆ 𝑦) → [(𝑧 ∩ 𝑤) / 𝑦]𝐵 ⊆ 𝑦)
37	36	a1i 11	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑧 ⊆ 𝐴 ∧ 𝑤 ⊆ 𝐴) → (([𝑧 / 𝑦]𝐵 ⊆ 𝑦 ∧ [𝑤 / 𝑦]𝐵 ⊆ 𝑦) → [(𝑧 ∩ 𝑤) / 𝑦]𝐵 ⊆ 𝑦))
38	6, 7, 12, 19, 29, 37	isfild 23806	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ 𝐵 ⊆ 𝑥} ∈ (Fil‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   ∈ wcel 2098   ≠ wne 2929  {crab 3418  [wsbc 3773   ∩ cin 3943   ⊆ wss 3944  ∅c0 4322  𝒫 cpw 4604  ‘cfv 6549  Filcfil 23793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557  df-fbas 21293  df-fil 23794

This theorem is referenced by:  fclscf  23973  flimfnfcls  23976