Theorem supfil 21509

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 3590	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑦))
2	1	elrab 3331	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝐵 ⊆ 𝑥} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ 𝐵 ⊆ 𝑦))
3		selpw 4115	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
4	3	anbi1i 727	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝐵 ⊆ 𝑦) ↔ (𝑦 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝑦))
5	2, 4	bitri 263	. . 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		elex 3185	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
8	7	3ad2ant1 1075	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V)
9		simp2 1055	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → 𝐵 ⊆ 𝐴)
10		sseq2 3590	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝐴))
11	10	sbcieg 3435	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝐴))
12	8, 11	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ([𝐴 / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝐴))
13	9, 12	mpbird 246	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → [𝐴 / 𝑦]𝐵 ⊆ 𝑦)
14		ss0 3926	. . . . 5 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅)
15	14	necon3ai 2807	. . . 4 ⊢ (𝐵 ≠ ∅ → ¬ 𝐵 ⊆ ∅)
16	15	3ad2ant3 1077	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ¬ 𝐵 ⊆ ∅)
17		0ex 4718	. . . 4 ⊢ ∅ ∈ V
18		sseq2 3590	. . . 4 ⊢ (𝑦 = ∅ → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ ∅))
19	17, 18	sbcie 3437	. . 3 ⊢ ([∅ / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ ∅)
20	16, 19	sylnibr 318	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ¬ [∅ / 𝑦]𝐵 ⊆ 𝑦)
21		sstr 3576	. . . . 5 ⊢ ((𝐵 ⊆ 𝑤 ∧ 𝑤 ⊆ 𝑧) → 𝐵 ⊆ 𝑧)
22	21	expcom 450	. . . 4 ⊢ (𝑤 ⊆ 𝑧 → (𝐵 ⊆ 𝑤 → 𝐵 ⊆ 𝑧))
23		vex 3176	. . . . 5 ⊢ 𝑤 ∈ V
24		sseq2 3590	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑤))
25	23, 24	sbcie 3437	. . . 4 ⊢ ([𝑤 / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑤)
26		vex 3176	. . . . 5 ⊢ 𝑧 ∈ V
27		sseq2 3590	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑧))
28	26, 27	sbcie 3437	. . . 4 ⊢ ([𝑧 / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ 𝑧)
29	22, 25, 28	3imtr4g 284	. . 3 ⊢ (𝑤 ⊆ 𝑧 → ([𝑤 / 𝑦]𝐵 ⊆ 𝑦 → [𝑧 / 𝑦]𝐵 ⊆ 𝑦))
30	29	3ad2ant3 1077	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑧 ⊆ 𝐴 ∧ 𝑤 ⊆ 𝑧) → ([𝑤 / 𝑦]𝐵 ⊆ 𝑦 → [𝑧 / 𝑦]𝐵 ⊆ 𝑦))
31		ssin 3797	. . . . . 6 ⊢ ((𝐵 ⊆ 𝑧 ∧ 𝐵 ⊆ 𝑤) ↔ 𝐵 ⊆ (𝑧 ∩ 𝑤))
32	31	biimpi 205	. . . . 5 ⊢ ((𝐵 ⊆ 𝑧 ∧ 𝐵 ⊆ 𝑤) → 𝐵 ⊆ (𝑧 ∩ 𝑤))
33	28, 25, 32	syl2anb 495	. . . 4 ⊢ (([𝑧 / 𝑦]𝐵 ⊆ 𝑦 ∧ [𝑤 / 𝑦]𝐵 ⊆ 𝑦) → 𝐵 ⊆ (𝑧 ∩ 𝑤))
34	26	inex1 4727	. . . . 5 ⊢ (𝑧 ∩ 𝑤) ∈ V
35		sseq2 3590	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ (𝑧 ∩ 𝑤)))
36	34, 35	sbcie 3437	. . . 4 ⊢ ([(𝑧 ∩ 𝑤) / 𝑦]𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ (𝑧 ∩ 𝑤))
37	33, 36	sylibr 223	. . 3 ⊢ (([𝑧 / 𝑦]𝐵 ⊆ 𝑦 ∧ [𝑤 / 𝑦]𝐵 ⊆ 𝑦) → [(𝑧 ∩ 𝑤) / 𝑦]𝐵 ⊆ 𝑦)
38	37	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))
39	6, 8, 13, 20, 30, 38	isfild 21472	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ 𝐵 ⊆ 𝑥} ∈ (Fil‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173  [wsbc 3402   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ‘cfv 5804  Filcfil 21459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-fbas 19564  df-fil 21460

This theorem is referenced by:  fclscf  21639  flimfnfcls  21642