Theorem elflim2 23920

Description: The predicate "is a limit point of a filter." (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
flimval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
elflim2	⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))

Proof of Theorem elflim2

Dummy variables 𝑥 𝑓 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		anass 468	. 2 ⊢ ((((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))))
2		df-3an 1089	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝐹 ⊆ 𝒫 𝑋))
3	2	anbi1i 625	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)) ↔ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
4		df-flim 23895	. . . 4 ⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
5	4	elmpocl 7609	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil))
6		flimval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
7	6	flimval 23919	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fLim 𝐹) = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)})
8	7	eleq2d 2823	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)}))
9		sneq 4592	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
10	9	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝐴}))
11	10	sseq1d 3967	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ↔ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
12	11	anbi1d 632	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)))
13	12	biancomd 463	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
14	13	elrab 3648	. . . . 5 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)} ↔ (𝐴 ∈ 𝑋 ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
15		an12 646	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
16	14, 15	bitri 275	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)} ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
17	8, 16	bitrdi 287	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))))
18	5, 17	biadanii 822	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))))
19	1, 3, 18	3bitr4ri 304	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {crab 3401   ⊆ wss 3903  𝒫 cpw 4556  {csn 4582  ∪ cuni 4865  ran crn 5633  ‘cfv 6500  (class class class)co 7368  Topctop 22849  neicnei 23053  Filcfil 23801   fLim cflim 23890

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-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-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  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-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-top 22850  df-flim 23895

This theorem is referenced by:  flimtop  23921  flimneiss  23922  flimelbas  23924  flimfil  23925  elflim  23927