Theorem flimopn 24022

Description: The condition for being a limit point of a filter still holds if one only considers open neighborhoods. (Contributed by Jeff Hankins, 4-Sep-2009.) (Proof shortened by Mario Carneiro, 9-Apr-2015.)

Assertion

Ref	Expression
flimopn	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐽   𝑥,𝑋

Proof of Theorem flimopn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elflim 24018	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
2		dfss3 3923	. . . 4 ⊢ (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦 ∈ 𝐹)
3		topontop 22960	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4	3	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
5		opnneip 23166	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝐴 ∈ 𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
6	5	3expb 1132	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝐴 ∈ 𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
7	4, 6	sylan 589	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝐽 ∧ 𝐴 ∈ 𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
8		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐹 ↔ 𝑥 ∈ 𝐹))
9	8	rspcv 3576	. . . . . . . . 9 ⊢ (𝑥 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦 ∈ 𝐹 → 𝑥 ∈ 𝐹))
10	7, 9	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝐽 ∧ 𝐴 ∈ 𝑥)) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦 ∈ 𝐹 → 𝑥 ∈ 𝐹))
11	10	expr 460	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝐴 ∈ 𝑥 → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦 ∈ 𝐹 → 𝑥 ∈ 𝐹)))
12	11	com23 86	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝐽) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦 ∈ 𝐹 → (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹)))
13	12	ralrimdva 3161	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦 ∈ 𝐹 → ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹)))
14		simpr 488	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
15	3	ad3antrrr 740	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
16		simplr 778	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ 𝑋)
17		toponuni 22961	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
18	17	ad3antrrr 740	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑋 = ∪ 𝐽)
19	16, 18	eleqtrd 2863	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ ∪ 𝐽)
20	19	snssd 4742	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ∪ 𝐽)
21		eqid 2761	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
22	21	neii1 23153	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 ⊆ ∪ 𝐽)
23	4, 22	sylan 589	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 ⊆ ∪ 𝐽)
24	21	neiint 23151	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑦 ⊆ ∪ 𝐽) → (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
25	15, 20, 23, 24	syl3anc 1389	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
26	14, 25	mpbid 234	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ((int‘𝐽)‘𝑦))
27		snssg 4739	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ ((int‘𝐽)‘𝑦) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
28	27	ad2antlr 737	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (𝐴 ∈ ((int‘𝐽)‘𝑦) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
29	26, 28	mpbird 259	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ ((int‘𝐽)‘𝑦))
30	21	ntropn 23096	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑦) ∈ 𝐽)
31	15, 23, 30	syl2anc 593	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑦) ∈ 𝐽)
32		eleq2 2850	. . . . . . . . . . 11 ⊢ (𝑥 = ((int‘𝐽)‘𝑦) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ((int‘𝐽)‘𝑦)))
33		eleq1 2849	. . . . . . . . . . 11 ⊢ (𝑥 = ((int‘𝐽)‘𝑦) → (𝑥 ∈ 𝐹 ↔ ((int‘𝐽)‘𝑦) ∈ 𝐹))
34	32, 33	imbi12d 346	. . . . . . . . . 10 ⊢ (𝑥 = ((int‘𝐽)‘𝑦) → ((𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹) ↔ (𝐴 ∈ ((int‘𝐽)‘𝑦) → ((int‘𝐽)‘𝑦) ∈ 𝐹)))
35	34	rspcv 3576	. . . . . . . . 9 ⊢ (((int‘𝐽)‘𝑦) ∈ 𝐽 → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹) → (𝐴 ∈ ((int‘𝐽)‘𝑦) → ((int‘𝐽)‘𝑦) ∈ 𝐹)))
36	31, 35	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹) → (𝐴 ∈ ((int‘𝐽)‘𝑦) → ((int‘𝐽)‘𝑦) ∈ 𝐹)))
37	29, 36	mpid 44	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹) → ((int‘𝐽)‘𝑦) ∈ 𝐹))
38		simpllr 785	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐹 ∈ (Fil‘𝑋))
39	21	ntrss2 23104	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑦) ⊆ 𝑦)
40	15, 23, 39	syl2anc 593	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑦) ⊆ 𝑦)
41	23, 18	sseqtrrd 3971	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 ⊆ 𝑋)
42		filss 23900	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (((int‘𝐽)‘𝑦) ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑦) ⊆ 𝑦)) → 𝑦 ∈ 𝐹)
43	42	3exp2 1367	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → (((int‘𝐽)‘𝑦) ∈ 𝐹 → (𝑦 ⊆ 𝑋 → (((int‘𝐽)‘𝑦) ⊆ 𝑦 → 𝑦 ∈ 𝐹))))
44	43	com24 95	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → (((int‘𝐽)‘𝑦) ⊆ 𝑦 → (𝑦 ⊆ 𝑋 → (((int‘𝐽)‘𝑦) ∈ 𝐹 → 𝑦 ∈ 𝐹))))
45	38, 40, 41, 44	syl3c 66	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (((int‘𝐽)‘𝑦) ∈ 𝐹 → 𝑦 ∈ 𝐹))
46	37, 45	syld 47	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹) → 𝑦 ∈ 𝐹))
47	46	ralrimdva 3161	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦 ∈ 𝐹))
48	13, 47	impbid 214	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦 ∈ 𝐹 ↔ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹)))
49	2, 48	bitrid 285	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ↔ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹)))
50	49	pm5.32da 587	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹))))
51	1, 50	bitrd 281	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ⊆ wss 3902  {csn 4579  ∪ cuni 4862  ‘cfv 6515  (class class class)co 7390  Topctop 22940  TopOnctopon 22957  intcnt 23064  neicnei 23144  Filcfil 23892   fLim cflim 23981

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-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

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-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  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-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-fbas 21408  df-top 22941  df-topon 22958  df-ntr 23067  df-nei 23145  df-fil 23893  df-flim 23986

This theorem is referenced by:  fbflim  24023  flimrest  24030  flimsncls  24033  isflf  24040  cnpflfi  24046  flimfnfcls  24075  alexsublem  24091  cfilfcls  25323  iscmet3lem2  25341