Theorem supfil 22500

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 3941	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑦))
2	1	elrab 3628	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝐵 ⊆ 𝑥} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ 𝐵 ⊆ 𝑦))
3		velpw 4502	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
4	3	anbi1i 626	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝐵 ⊆ 𝑦) ↔ (𝑦 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝑦))
5	2, 4	bitri 278	. . 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 3941	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝐴))
10	9	sbcieg 3758	. . . 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 260	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → [𝐴 / 𝑦]𝐵 ⊆ 𝑦)
13		ss0 4306	. . . . 5 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅)
14	13	necon3ai 3012	. . . 4 ⊢ (𝐵 ≠ ∅ → ¬ 𝐵 ⊆ ∅)
15	14	3ad2ant3 1132	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ¬ 𝐵 ⊆ ∅)
16		0ex 5175	. . . 4 ⊢ ∅ ∈ V
17		sseq2 3941	. . . 4 ⊢ (𝑦 = ∅ → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ ∅))
18	16, 17	sbcie 3760	. . 3 ⊢ ([∅ / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ ∅)
19	15, 18	sylnibr 332	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ¬ [∅ / 𝑦]𝐵 ⊆ 𝑦)
20		sstr 3923	. . . . 5 ⊢ ((𝐵 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑧) → 𝐵 ⊆ 𝑧)
21	20	expcom 417	. . . 4 ⊢ (𝑤 ⊆ 𝑧 → (𝐵 ⊆ 𝑤 → 𝐵 ⊆ 𝑧))
22		vex 3444	. . . . 5 ⊢ 𝑤 ∈ V
23		sseq2 3941	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑤))
24	22, 23	sbcie 3760	. . . 4 ⊢ ([𝑤 / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑤)
25		vex 3444	. . . . 5 ⊢ 𝑧 ∈ V
26		sseq2 3941	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑧))
27	25, 26	sbcie 3760	. . . 4 ⊢ ([𝑧 / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑧)
28	21, 24, 27	3imtr4g 299	. . 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 4157	. . . . . 6 ⊢ ((𝐵 ⊆ 𝑧 ∧ 𝐵 ⊆ 𝑤) ↔ 𝐵 ⊆ (𝑧 ∩ 𝑤))
31	30	biimpi 219	. . . . 5 ⊢ ((𝐵 ⊆ 𝑧 ∧ 𝐵 ⊆ 𝑤) → 𝐵 ⊆ (𝑧 ∩ 𝑤))
32	27, 24, 31	syl2anb 600	. . . 4 ⊢ (([𝑧 / 𝑦]𝐵 ⊆ 𝑦 ∧ [𝑤 / 𝑦]𝐵 ⊆ 𝑦) → 𝐵 ⊆ (𝑧 ∩ 𝑤))
33	25	inex1 5185	. . . . 5 ⊢ (𝑧 ∩ 𝑤) ∈ V
34		sseq2 3941	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ (𝑧 ∩ 𝑤)))
35	33, 34	sbcie 3760	. . . 4 ⊢ ([(𝑧 ∩ 𝑤) / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ (𝑧 ∩ 𝑤))
36	32, 35	sylibr 237	. . 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 22463	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ 𝐵 ⊆ 𝑥} ∈ (Fil‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111   ≠ wne 2987  {crab 3110  [wsbc 3720   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  ‘cfv 6324  Filcfil 22450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-fbas 20088  df-fil 22451

This theorem is referenced by:  fclscf  22630  flimfnfcls  22633