Theorem flftg 23883

Description: Limit points of a function can be defined using topological bases. (Contributed by Mario Carneiro, 19-Sep-2015.)

Hypothesis

Ref	Expression
flftg.l	⊢ 𝐽 = (topGen‘𝐵)

Assertion

Ref	Expression
flftg	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))))

Distinct variable groups:   𝑜,𝑠,𝐴   𝐵,𝑜   𝑜,𝐹,𝑠   𝐽,𝑠   𝑜,𝐿,𝑠   𝑋,𝑠   𝑌,𝑠

Allowed substitution hints:   𝐵(𝑠)   𝐽(𝑜)   𝑋(𝑜)   𝑌(𝑜)

Proof of Theorem flftg

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isflf 23880	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))))
2		flftg.l	. . . . 5 ⊢ 𝐽 = (topGen‘𝐵)
3	2	raleqi 3297	. . . 4 ⊢ (∀𝑢 ∈ 𝐽 (𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ ∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
4		simpl1 1192	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
5		topontop 22800	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
6	4, 5	syl 17	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
7	2, 6	eqeltrrid 2833	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (topGen‘𝐵) ∈ Top)
8		tgclb 22857	. . . . . . 7 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
9	7, 8	sylibr 234	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐵 ∈ TopBases)
10		bastg 22853	. . . . . 6 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
11		eleq2w 2812	. . . . . . . . 9 ⊢ (𝑢 = 𝑜 → (𝐴 ∈ 𝑢 ↔ 𝐴 ∈ 𝑜))
12		sseq2 3973	. . . . . . . . . 10 ⊢ (𝑢 = 𝑜 → ((𝐹 “ 𝑠) ⊆ 𝑢 ↔ (𝐹 “ 𝑠) ⊆ 𝑜))
13	12	rexbidv 3157	. . . . . . . . 9 ⊢ (𝑢 = 𝑜 → (∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢 ↔ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))
14	11, 13	imbi12d 344	. . . . . . . 8 ⊢ (𝑢 = 𝑜 → ((𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
15	14	cbvralvw 3215	. . . . . . 7 ⊢ (∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ ∀𝑜 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))
16		ssralv 4015	. . . . . . 7 ⊢ (𝐵 ⊆ (topGen‘𝐵) → (∀𝑜 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) → ∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
17	15, 16	biimtrid 242	. . . . . 6 ⊢ (𝐵 ⊆ (topGen‘𝐵) → (∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) → ∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
18	9, 10, 17	3syl 18	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) → ∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
19		tg2 22852	. . . . . . . 8 ⊢ ((𝑢 ∈ (topGen‘𝐵) ∧ 𝐴 ∈ 𝑢) → ∃𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢))
20		r19.29 3094	. . . . . . . . . 10 ⊢ ((∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ ∃𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢)) → ∃𝑜 ∈ 𝐵 ((𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢)))
21		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → 𝐴 ∈ 𝑜)
22		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → 𝑜 ⊆ 𝑢)
23		sstr2 3953	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 “ 𝑠) ⊆ 𝑜 → (𝑜 ⊆ 𝑢 → (𝐹 “ 𝑠) ⊆ 𝑢))
24	22, 23	syl5com 31	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → ((𝐹 “ 𝑠) ⊆ 𝑜 → (𝐹 “ 𝑠) ⊆ 𝑢))
25	24	reximdv 3148	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → (∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
26	21, 25	embantd 59	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → ((𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
27	26	impcom 407	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢)) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢)
28	27	rexlimivw 3130	. . . . . . . . . 10 ⊢ (∃𝑜 ∈ 𝐵 ((𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢)) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢)
29	20, 28	syl 17	. . . . . . . . 9 ⊢ ((∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ ∃𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢)) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢)
30	29	ex 412	. . . . . . . 8 ⊢ (∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) → (∃𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
31	19, 30	syl5 34	. . . . . . 7 ⊢ (∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) → ((𝑢 ∈ (topGen‘𝐵) ∧ 𝐴 ∈ 𝑢) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
32	31	expdimp 452	. . . . . 6 ⊢ ((∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ 𝑢 ∈ (topGen‘𝐵)) → (𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
33	32	ralrimiva 3125	. . . . 5 ⊢ (∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) → ∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
34	18, 33	impbid1 225	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ ∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
35	3, 34	bitrid 283	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (∀𝑢 ∈ 𝐽 (𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ ∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
36	35	pm5.32da 579	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐴 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))))
37	1, 36	bitrd 279	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914   “ cima 5641  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  topGenctg 17400  Topctop 22780  TopOnctopon 22797  TopBasesctb 22832  Filcfil 23732   fLimf cflf 23822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-topgen 17406  df-fbas 21261  df-fg 21262  df-top 22781  df-topon 22798  df-bases 22833  df-ntr 22907  df-nei 22985  df-fil 23733  df-fm 23825  df-flim 23826  df-flf 23827

This theorem is referenced by:  txflf  23893