Theorem elflim2 21987

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 459	. 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 1073	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝐹 ⊆ 𝒫 𝑋))
3	2	anbi1i 602	. 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 21962	. . . 4 ⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
5	4	elmpt2cl 7022	. . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil))
6		flimval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
7	6	flimval 21986	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fLim 𝐹) = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)})
8	7	eleq2d 2836	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)}))
9		sneq 4326	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
10	9	fveq2d 6336	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝐴}))
11	10	sseq1d 3781	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ↔ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
12	11	anbi1d 607	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)))
13		ancom 452	. . . . . . 7 ⊢ ((((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))
14	12, 13	syl6bb 276	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
15	14	elrab 3515	. . . . 5 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)} ↔ (𝐴 ∈ 𝑋 ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
16		an12 616	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
17	15, 16	bitri 264	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)} ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
18	8, 17	syl6bb 276	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))))
19	5, 18	biadan2 802	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))))
20	1, 3, 19	3bitr4ri 293	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))

Syntax hints:   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  {crab 3065   ⊆ wss 3723  𝒫 cpw 4297  {csn 4316  ∪ cuni 4574  ran crn 5250  ‘cfv 6031  (class class class)co 6792  Topctop 20917  neicnei 21121  Filcfil 21868   fLim cflim 21957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-top 20918  df-flim 21962

This theorem is referenced by:  flimtop  21988  flimneiss  21989  flimelbas  21991  flimfil  21992  elflim  21994