Theorem elflim2 24012

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 472	. 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 1099	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝐹 ⊆ 𝒫 𝑋))
3	2	anbi1i 633	. 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 23987	. . . 4 ⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
5	4	elmpocl 7632	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil))
6		flimval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
7	6	flimval 24011	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fLim 𝐹) = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)})
8	7	eleq2d 2847	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)}))
9		sneq 4589	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
10	9	fveq2d 6866	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝐴}))
11	10	sseq1d 3965	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ↔ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
12	11	anbi1d 640	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)))
13	12	biancomd 467	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
14	13	elrab 3649	. . . . 5 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)} ↔ (𝐴 ∈ 𝑋 ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
15		an12 655	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
16	14, 15	bitri 277	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)} ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
17	8, 16	bitrdi 289	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))))
18	5, 17	biadanii 831	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))))
19	1, 3, 18	3bitr4ri 306	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {crab 3413   ⊆ wss 3902  𝒫 cpw 4552  {csn 4579  ∪ cuni 4862  ran crn 5644  ‘cfv 6516  (class class class)co 7391  Topctop 22941  neicnei 23145  Filcfil 23893   fLim cflim 23982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-top 22942  df-flim 23987

This theorem is referenced by:  flimtop  24013  flimneiss  24014  flimelbas  24016  flimfil  24017  elflim  24019