Theorem fixufil 22532

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 22531	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}})))
2	1	simprd 498	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}}))
3	1	simpld 497	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {{𝐴}} ∈ (fBas‘𝑋))
4		fgcl 22488	. . . 4 ⊢ ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
5	3, 4	syl 17	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
6	2, 5	eqeltrd 2915	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋))
7		undif2 4427	. . . . . . . . . 10 ⊢ (𝑦 ∪ (𝑋 ∖ 𝑦)) = (𝑦 ∪ 𝑋)
8		elpwi 4550	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
9		ssequn1 4158	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑋 ↔ (𝑦 ∪ 𝑋) = 𝑋)
10	8, 9	sylib 220	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝑦 ∪ 𝑋) = 𝑋)
11	7, 10	syl5req 2871	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑋 = (𝑦 ∪ (𝑋 ∖ 𝑦)))
12	11	eleq2d 2900	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦))))
13	12	biimpac 481	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝒫 𝑋) → 𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦)))
14		elun 4127	. . . . . . 7 ⊢ (𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦)) ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
15	13, 14	sylib 220	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
16	15	adantll 712	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
17		ibar 531	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝐴 ∈ 𝑦 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦)))
18	17	adantl 484	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦)))
19		difss 4110	. . . . . . . . 9 ⊢ (𝑋 ∖ 𝑦) ⊆ 𝑋
20		elpw2g 5249	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑋))
21	19, 20	mpbiri 260	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
22	21	ad2antrr 724	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
23	22	biantrurd 535	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ (𝑋 ∖ 𝑦) ↔ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
24	18, 23	orbi12d 915	. . . . 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 234	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
26	25	ralrimiva 3184	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
27		eleq2 2903	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
28	27	elrab 3682	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦))
29		eleq2 2903	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ 𝑦) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (𝑋 ∖ 𝑦)))
30	29	elrab 3682	. . . . 5 ⊢ ((𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦)))
31	28, 30	orbi12i 911	. . . 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 3167	. . 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 236	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∨ (𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))
34		isufil 22513	. 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 585	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (UFil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  ∀wral 3140  {crab 3144   ∖ cdif 3935   ∪ cun 3936   ⊆ wss 3938  𝒫 cpw 4541  {csn 4569  ‘cfv 6357  (class class class)co 7158  fBascfbas 20535  filGencfg 20536  Filcfil 22455  UFilcufil 22509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-int 4879  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-fbas 20544  df-fg 20545  df-fil 22456  df-ufil 22511

This theorem is referenced by: (None)