Theorem elflim 23954

Description: The predicate "is a limit point of a filter." (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
elflim	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))

Proof of Theorem elflim

Step	Hyp	Ref	Expression
1		topontop 22896	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	adantr 481	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐽 ∈ Top)
3		fvssunirn 6858	. . . . 5 ⊢ (Fil‘𝑋) ⊆ ∪ ran Fil
4	3	sseli 3911	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ ∪ ran Fil)
5	4	adantl 482	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ ∪ ran Fil)
6		filsspw 23834	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
7	6	adantl 482	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
8		toponuni 22897	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
9	8	adantr 481	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝑋 = ∪ 𝐽)
10	9	pweqd 4546	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
11	7, 10	sseqtrd 3951	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 ∪ 𝐽)
12		eqid 2739	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
13	12	elflim2 23947	. . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽) ∧ (𝐴 ∈ ∪ 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
14	13	baib 540	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
15	2, 5, 11, 14	syl3anc 1379	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
16	9	eleq2d 2825	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ 𝐽))
17	16	anbi1d 637	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
18	15, 17	bitr4d 283	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  𝒫 cpw 4529  {csn 4555  ∪ cuni 4838  ran crn 5619  ‘cfv 6485  (class class class)co 7356  Topctop 22876  TopOnctopon 22893  neicnei 23080  Filcfil 23828   fLim cflim 23917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-fbas 21344  df-top 22877  df-topon 22894  df-fil 23829  df-flim 23922

This theorem is referenced by:  flimss2  23955  flimss1  23956  neiflim  23957  flimopn  23958  hausflim  23964  flimclslem  23967  flfnei  23974  fclsfnflim  24010