Theorem fixufil 23878

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 23877	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}})))
2	1	simprd 495	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}}))
3	1	simpld 494	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {{𝐴}} ∈ (fBas‘𝑋))
4		fgcl 23834	. . . 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 2837	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋))
7		undif2 4431	. . . . . . . . . 10 ⊢ (𝑦 ∪ (𝑋 ∖ 𝑦)) = (𝑦 ∪ 𝑋)
8		elpwi 4563	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
9		ssequn1 4140	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑋 ↔ (𝑦 ∪ 𝑋) = 𝑋)
10	8, 9	sylib 218	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝑦 ∪ 𝑋) = 𝑋)
11	7, 10	eqtr2id 2785	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑋 = (𝑦 ∪ (𝑋 ∖ 𝑦)))
12	11	eleq2d 2823	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦))))
13	12	biimpac 478	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝒫 𝑋) → 𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦)))
14		elun 4107	. . . . . . 7 ⊢ (𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦)) ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
15	13, 14	sylib 218	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
16	15	adantll 715	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
17		ibar 528	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝐴 ∈ 𝑦 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦)))
18	17	adantl 481	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦)))
19		difss 4090	. . . . . . . . 9 ⊢ (𝑋 ∖ 𝑦) ⊆ 𝑋
20		elpw2g 5280	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑋))
21	19, 20	mpbiri 258	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
22	21	ad2antrr 727	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
23	22	biantrurd 532	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ (𝑋 ∖ 𝑦) ↔ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
24	18, 23	orbi12d 919	. . . . 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 232	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
26	25	ralrimiva 3130	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
27		eleq2 2826	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
28	27	elrab 3648	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦))
29		eleq2 2826	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ 𝑦) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (𝑋 ∖ 𝑦)))
30	29	elrab 3648	. . . . 5 ⊢ ((𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦)))
31	28, 30	orbi12i 915	. . . 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 3084	. . 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 234	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∨ (𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))
34		isufil 23859	. 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 584	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (UFil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  𝒫 cpw 4556  {csn 4582  ‘cfv 6500  (class class class)co 7368  fBascfbas 21309  filGencfg 21310  Filcfil 23801  UFilcufil 23855

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-fbas 21318  df-fg 21319  df-fil 23802  df-ufil 23857

This theorem is referenced by: (None)