Theorem flffbas 24347

Description: Limit points of a function can be defined using filter bases. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)

Hypothesis

Ref	Expression
flffbas.l	⊢ 𝐿 = (𝑌filGen𝐵)

Assertion

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

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

Proof of Theorem flffbas

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		flffbas.l	. . . 4 ⊢ 𝐿 = (𝑌filGen𝐵)
2		fgcl 24230	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
3	1, 2	eqeltrid 2868	. . 3 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
4		isflf 24345	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜))))
5	3, 4	syl3an2 1182	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜))))
6	1	eleq2i 2856	. . . . . . . 8 ⊢ (𝑡 ∈ 𝐿 ↔ 𝑡 ∈ (𝑌filGen𝐵))
7		elfg 24223	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑡 ∈ (𝑌filGen𝐵) ↔ (𝑡 ⊆ 𝑌 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
8	7	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑡 ∈ (𝑌filGen𝐵) ↔ (𝑡 ⊆ 𝑌 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
9		sstr2 3920	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 “ 𝑠) ⊆ (𝐹 “ 𝑡) → ((𝐹 “ 𝑡) ⊆ 𝑜 → (𝐹 “ 𝑠) ⊆ 𝑜))
10		imass2 6068	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ⊆ 𝑡 → (𝐹 “ 𝑠) ⊆ (𝐹 “ 𝑡))
11	9, 10	syl11 34	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 “ 𝑡) ⊆ 𝑜 → (𝑠 ⊆ 𝑡 → (𝐹 “ 𝑠) ⊆ 𝑜))
12	11	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐹 “ 𝑡) ⊆ 𝑜) → (𝑠 ⊆ 𝑡 → (𝐹 “ 𝑠) ⊆ 𝑜))
13	12	reximdv 3181	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐹 “ 𝑡) ⊆ 𝑜) → (∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜))
14	13	ex 418	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐹 “ 𝑡) ⊆ 𝑜 → (∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜)))
15	14	com23 87	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡 → ((𝐹 “ 𝑡) ⊆ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜)))
16	15	adantld 496	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑡 ⊆ 𝑌 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡) → ((𝐹 “ 𝑡) ⊆ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜)))
17	8, 16	sylbid 243	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑡 ∈ (𝑌filGen𝐵) → ((𝐹 “ 𝑡) ⊆ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜)))
18	17	adantr 486	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (𝑡 ∈ (𝑌filGen𝐵) → ((𝐹 “ 𝑡) ⊆ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜)))
19	6, 18	biimtrid 245	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (𝑡 ∈ 𝐿 → ((𝐹 “ 𝑡) ⊆ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜)))
20	19	rexlimdv 3165	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜))
21		ssfg 24224	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
22	21, 1	sseqtrrdi 3954	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ 𝐿)
23	22	sselda 3913	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ 𝐿)
24	23	3ad2antl2 1205	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ 𝐿)
25	24	ad2ant2r 760	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ (𝑠 ∈ 𝐵 ∧ (𝐹 “ 𝑠) ⊆ 𝑜)) → 𝑠 ∈ 𝐿)
26		simprr 785	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ (𝑠 ∈ 𝐵 ∧ (𝐹 “ 𝑠) ⊆ 𝑜)) → (𝐹 “ 𝑠) ⊆ 𝑜)
27		imaeq2 6061	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (𝐹 “ 𝑡) = (𝐹 “ 𝑠))
28	27	sseq1d 3944	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ((𝐹 “ 𝑡) ⊆ 𝑜 ↔ (𝐹 “ 𝑠) ⊆ 𝑜))
29	28	rspcev 3580	. . . . . . . 8 ⊢ ((𝑠 ∈ 𝐿 ∧ (𝐹 “ 𝑠) ⊆ 𝑜) → ∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜)
30	25, 26, 29	syl2anc 596	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ (𝑠 ∈ 𝐵 ∧ (𝐹 “ 𝑠) ⊆ 𝑜)) → ∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜)
31	30	rexlimdvaa 3168	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜 → ∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜))
32	20, 31	impbid 215	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜 ↔ ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜))
33	32	imbi2d 343	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝐴 ∈ 𝑜 → ∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜) ↔ (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜)))
34	33	ralbidv 3189	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜)))
35	34	pm5.32da 590	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜))))
36	5, 35	bitrd 282	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   “ cima 5659  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446  fBascfbas 21699  filGencfg 21700  TopOnctopon 23261  Filcfil 24197   fLimf cflf 24287

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-map 8880  df-fbas 21708  df-fg 21709  df-top 23245  df-topon 23262  df-ntr 23371  df-nei 23449  df-fil 24198  df-fm 24290  df-flim 24291  df-flf 24292

This theorem is used by:  lmflf  24357  eltsms  24485