Theorem supfil 23924

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 4035	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑦))
2	1	elrab 3708	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝐵 ⊆ 𝑥} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ 𝐵 ⊆ 𝑦))
3		velpw 4627	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
4	3	anbi1i 623	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝐵 ⊆ 𝑦) ↔ (𝑦 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝑦))
5	2, 4	bitri 275	. . 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 1136	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → 𝐴 ∈ 𝑉)
8		simp2 1137	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → 𝐵 ⊆ 𝐴)
9		sseq2 4035	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝐴))
10	9	sbcieg 3845	. . . 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 257	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → [𝐴 / 𝑦]𝐵 ⊆ 𝑦)
13		ss0 4425	. . . . 5 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅)
14	13	necon3ai 2971	. . . 4 ⊢ (𝐵 ≠ ∅ → ¬ 𝐵 ⊆ ∅)
15	14	3ad2ant3 1135	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ¬ 𝐵 ⊆ ∅)
16		0ex 5325	. . . 4 ⊢ ∅ ∈ V
17		sseq2 4035	. . . 4 ⊢ (𝑦 = ∅ → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ ∅))
18	16, 17	sbcie 3848	. . 3 ⊢ ([∅ / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ ∅)
19	15, 18	sylnibr 329	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ¬ [∅ / 𝑦]𝐵 ⊆ 𝑦)
20		sstr 4017	. . . . 5 ⊢ ((𝐵 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑧) → 𝐵 ⊆ 𝑧)
21	20	expcom 413	. . . 4 ⊢ (𝑤 ⊆ 𝑧 → (𝐵 ⊆ 𝑤 → 𝐵 ⊆ 𝑧))
22		vex 3492	. . . . 5 ⊢ 𝑤 ∈ V
23		sseq2 4035	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑤))
24	22, 23	sbcie 3848	. . . 4 ⊢ ([𝑤 / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑤)
25		vex 3492	. . . . 5 ⊢ 𝑧 ∈ V
26		sseq2 4035	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑧))
27	25, 26	sbcie 3848	. . . 4 ⊢ ([𝑧 / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑧)
28	21, 24, 27	3imtr4g 296	. . 3 ⊢ (𝑤 ⊆ 𝑧 → ([𝑤 / 𝑦]𝐵 ⊆ 𝑦 → [𝑧 / 𝑦]𝐵 ⊆ 𝑦))
29	28	3ad2ant3 1135	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑧 ⊆ 𝐴 ∧ 𝑤 ⊆ 𝑧) → ([𝑤 / 𝑦]𝐵 ⊆ 𝑦 → [𝑧 / 𝑦]𝐵 ⊆ 𝑦))
30		ssin 4260	. . . . . 6 ⊢ ((𝐵 ⊆ 𝑧 ∧ 𝐵 ⊆ 𝑤) ↔ 𝐵 ⊆ (𝑧 ∩ 𝑤))
31	30	biimpi 216	. . . . 5 ⊢ ((𝐵 ⊆ 𝑧 ∧ 𝐵 ⊆ 𝑤) → 𝐵 ⊆ (𝑧 ∩ 𝑤))
32	27, 24, 31	syl2anb 597	. . . 4 ⊢ (([𝑧 / 𝑦]𝐵 ⊆ 𝑦 ∧ [𝑤 / 𝑦]𝐵 ⊆ 𝑦) → 𝐵 ⊆ (𝑧 ∩ 𝑤))
33	25	inex1 5335	. . . . 5 ⊢ (𝑧 ∩ 𝑤) ∈ V
34		sseq2 4035	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ (𝑧 ∩ 𝑤)))
35	33, 34	sbcie 3848	. . . 4 ⊢ ([(𝑧 ∩ 𝑤) / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ (𝑧 ∩ 𝑤))
36	32, 35	sylibr 234	. . 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 23887	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ 𝐵 ⊆ 𝑥} ∈ (Fil‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108   ≠ wne 2946  {crab 3443  [wsbc 3804   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  ‘cfv 6573  Filcfil 23874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-fbas 21384  df-fil 23875

This theorem is referenced by:  fclscf  24054  flimfnfcls  24057