Theorem elflim2 23988

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 1088	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝐹 ⊆ 𝒫 𝑋))
3	2	anbi1i 624	. 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 23963	. . . 4 ⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
5	4	elmpocl 7674	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil))
6		flimval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
7	6	flimval 23987	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fLim 𝐹) = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)})
8	7	eleq2d 2825	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)}))
9		sneq 4641	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
10	9	fveq2d 6911	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝐴}))
11	10	sseq1d 4027	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ↔ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
12	11	anbi1d 631	. . . . . . 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 3695	. . . . 5 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)} ↔ (𝐴 ∈ 𝑋 ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
15		an12 645	. . . . 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 1086   = wceq 1537   ∈ wcel 2106  {crab 3433   ⊆ wss 3963  𝒫 cpw 4605  {csn 4631  ∪ cuni 4912  ran crn 5690  ‘cfv 6563  (class class class)co 7431  Topctop 22915  neicnei 23121  Filcfil 23869   fLim cflim 23958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-top 22916  df-flim 23963

This theorem is referenced by:  flimtop  23989  flimneiss  23990  flimelbas  23992  flimfil  23993  elflim  23995