Theorem flftg 23969

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 23966	. 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 3308	. . . 4 ⊢ (∀𝑢 ∈ 𝐽 (𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ ∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
4		simpl1 1191	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
5		topontop 22886	. . . . . . . . 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 2838	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (topGen‘𝐵) ∈ Top)
8		tgclb 22943	. . . . . . 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 22939	. . . . . 6 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
11		eleq2w 2817	. . . . . . . . 9 ⊢ (𝑢 = 𝑜 → (𝐴 ∈ 𝑢 ↔ 𝐴 ∈ 𝑜))
12		sseq2 3992	. . . . . . . . . 10 ⊢ (𝑢 = 𝑜 → ((𝐹 “ 𝑠) ⊆ 𝑢 ↔ (𝐹 “ 𝑠) ⊆ 𝑜))
13	12	rexbidv 3166	. . . . . . . . 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 3224	. . . . . . 7 ⊢ (∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ ∀𝑜 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))
16		ssralv 4034	. . . . . . 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 22938	. . . . . . . 8 ⊢ ((𝑢 ∈ (topGen‘𝐵) ∧ 𝐴 ∈ 𝑢) → ∃𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢))
20		r19.29 3101	. . . . . . . . . 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 3972	. . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . 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 3138	. . . . . . . . . 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 3133	. . . . 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 1539   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059   ⊆ wss 3933   “ cima 5670  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414  topGenctg 17458  Topctop 22866  TopOnctopon 22883  TopBasesctb 22918  Filcfil 23818   fLimf cflf 23908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8851  df-topgen 17464  df-fbas 21328  df-fg 21329  df-top 22867  df-topon 22884  df-bases 22919  df-ntr 22993  df-nei 23071  df-fil 23819  df-fm 23911  df-flim 23912  df-flf 23913

This theorem is referenced by:  txflf  23979