Theorem fixufil 24221

Description: The condition describing a fixed ultrafilter always produces an ultrafilter. (Contributed by Jeff Hankins, 9-Dec-2009.) (Revised by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 29-Jul-2015.)

Assertion

Ref	Expression
fixufil	⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (UFil‘𝑋))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝑥,𝑉

Proof of Theorem fixufil

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uffix 24220	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}})))
2	1	simprd 501	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}}))
3	1	simpld 500	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {{𝐴}} ∈ (fBas‘𝑋))
4		fgcl 24177	. . . 4 ⊢ ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
5	3, 4	syl 18	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
6	2, 5	eqeltrd 2861	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋))
7		undif2 4431	. . . . . . . . . 10 ⊢ (𝑦 ∪ (𝑋 ∖ 𝑦)) = (𝑦 ∪ 𝑋)
8		elpwi 4564	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
9		ssequn1 4132	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑋 ↔ (𝑦 ∪ 𝑋) = 𝑋)
10	8, 9	sylib 221	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝑦 ∪ 𝑋) = 𝑋)
11	7, 10	eqtr2id 2809	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑋 = (𝑦 ∪ (𝑋 ∖ 𝑦)))
12	11	eleq2d 2847	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦))))
13	12	biimpac 484	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝒫 𝑋) → 𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦)))
14		elun 4100	. . . . . . 7 ⊢ (𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦)) ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
15	13, 14	sylib 221	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
16	15	adantll 727	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
17		ibar 538	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝐴 ∈ 𝑦 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦)))
18	17	adantl 487	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦)))
19		difss 4083	. . . . . . . . 9 ⊢ (𝑋 ∖ 𝑦) ⊆ 𝑋
20		elpw2g 5295	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑋))
21	19, 20	mpbiri 261	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
22	21	ad2antrr 739	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
23	22	biantrurd 542	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ (𝑋 ∖ 𝑦) ↔ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
24	18, 23	orbi12d 932	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)) ↔ ((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦)))))
25	16, 24	mpbid 235	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
26	25	ralrimiva 3155	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
27		eleq2 2850	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
28	27	elrab 3645	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦))
29		eleq2 2850	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ 𝑦) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (𝑋 ∖ 𝑦)))
30	29	elrab 3645	. . . . 5 ⊢ ((𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦)))
31	28, 30	orbi12i 928	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∨ (𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) ↔ ((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
32	31	ralbii 3109	. . 3 ⊢ (∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∨ (𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) ↔ ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
33	26, 32	sylibr 237	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∨ (𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))
34		isufil 24202	. 2 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (UFil‘𝑋) ↔ ({𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋) ∧ ∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∨ (𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})))
35	6, 33, 34	sylanbrc 595	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (UFil‘𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584  ‘cfv 6531  (class class class)co 7412  fBascfbas 21646  filGencfg 21647  Filcfil 24144  UFilcufil 24198

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-fbas 21655  df-fg 21656  df-fil 24145  df-ufil 24200

This theorem is used by: (None)