Theorem flftg 24228

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 24225	. 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 3319	. . . 4 ⊢ (∀𝑢 ∈ 𝐽 (𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ ∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
4		simpl1 1210	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
5		topontop 23144	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
6	4, 5	syl 18	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
7	2, 6	eqeltrrid 2867	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (topGen‘𝐵) ∈ Top)
8		tgclb 23201	. . . . . . 7 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
9	7, 8	sylibr 237	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐵 ∈ TopBases)
10		bastg 23197	. . . . . 6 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
11		eleq2w 2846	. . . . . . . . 9 ⊢ (𝑢 = 𝑜 → (𝐴 ∈ 𝑢 ↔ 𝐴 ∈ 𝑜))
12		sseq2 3960	. . . . . . . . . 10 ⊢ (𝑢 = 𝑜 → ((𝐹 “ 𝑠) ⊆ 𝑢 ↔ (𝐹 “ 𝑠) ⊆ 𝑜))
13	12	rexbidv 3188	. . . . . . . . 9 ⊢ (𝑢 = 𝑜 → (∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢 ↔ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))
14	11, 13	imbi12d 347	. . . . . . . 8 ⊢ (𝑢 = 𝑜 → ((𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
15	14	cbvralvw 3242	. . . . . . 7 ⊢ (∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ ∀𝑜 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))
16		ssralv 4003	. . . . . . 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 245	. . . . . 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 19	. . . . 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 23196	. . . . . . . 8 ⊢ ((𝑢 ∈ (topGen‘𝐵) ∧ 𝐴 ∈ 𝑢) → ∃𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢))
20		r19.29 3127	. . . . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → 𝐴 ∈ 𝑜)
22		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → 𝑜 ⊆ 𝑢)
23		sstr2 3941	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 “ 𝑠) ⊆ 𝑜 → (𝑜 ⊆ 𝑢 → (𝐹 “ 𝑠) ⊆ 𝑢))
24	22, 23	syl5com 32	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → ((𝐹 “ 𝑠) ⊆ 𝑜 → (𝐹 “ 𝑠) ⊆ 𝑢))
25	24	reximdv 3179	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → (∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
26	21, 25	embantd 60	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → ((𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
27	26	impcom 413	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢)) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢)
28	27	rexlimivw 3161	. . . . . . . . . 10 ⊢ (∃𝑜 ∈ 𝐵 ((𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢)) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢)
29	20, 28	syl 18	. . . . . . . . 9 ⊢ ((∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ ∃𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢)) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢)
30	29	ex 418	. . . . . . . 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 35	. . . . . . 7 ⊢ (∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) → ((𝑢 ∈ (topGen‘𝐵) ∧ 𝐴 ∈ 𝑢) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
32	31	expdimp 458	. . . . . 6 ⊢ ((∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ 𝑢 ∈ (topGen‘𝐵)) → (𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
33	32	ralrimiva 3156	. . . . 5 ⊢ (∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) → ∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
34	18, 33	impbid1 228	. . . 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 286	. . 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 590	. 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 282	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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088   ⊆ wss 3902   “ cima 5662  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417  topGenctg 17528  Topctop 23124  TopOnctopon 23141  TopBasesctb 23176  Filcfil 24077   fLimf cflf 24167

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-map 8832  df-topgen 17534  df-fbas 21588  df-fg 21589  df-top 23125  df-topon 23142  df-bases 23177  df-ntr 23251  df-nei 23329  df-fil 24078  df-fm 24170  df-flim 24171  df-flf 24172

This theorem is used by:  txflf  24238