Theorem flffbas 24113

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 23996	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
3	1, 2	eqeltrid 2869	. . 3 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
4		isflf 24111	. . 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 1180	. 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 2857	. . . . . . . 8 ⊢ (𝑡 ∈ 𝐿 ↔ 𝑡 ∈ (𝑌filGen𝐵))
7		elfg 23989	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑡 ∈ (𝑌filGen𝐵) ↔ (𝑡 ⊆ 𝑌 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
8	7	3ad2ant2 1150	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑡 ∈ (𝑌filGen𝐵) ↔ (𝑡 ⊆ 𝑌 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
9		sstr2 3946	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 “ 𝑠) ⊆ (𝐹 “ 𝑡) → ((𝐹 “ 𝑡) ⊆ 𝑜 → (𝐹 “ 𝑠) ⊆ 𝑜))
10		imass2 6095	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ⊆ 𝑡 → (𝐹 “ 𝑠) ⊆ (𝐹 “ 𝑡))
11	9, 10	syl11 34	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 “ 𝑡) ⊆ 𝑜 → (𝑠 ⊆ 𝑡 → (𝐹 “ 𝑠) ⊆ 𝑜))
12	11	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐹 “ 𝑡) ⊆ 𝑜) → (𝑠 ⊆ 𝑡 → (𝐹 “ 𝑠) ⊆ 𝑜))
13	12	reximdv 3180	. . . . . . . . . . . . 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 417	. . . . . . . . . . . 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 495	. . . . . . . . . 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 485	. . . . . . . 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 3164	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜))
21		ssfg 23990	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
22	21, 1	sseqtrrdi 3980	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ 𝐿)
23	22	sselda 3939	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ 𝐿)
24	23	3ad2antl2 1203	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ 𝐿)
25	24	ad2ant2r 759	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ (𝑠 ∈ 𝐵 ∧ (𝐹 “ 𝑠) ⊆ 𝑜)) → 𝑠 ∈ 𝐿)
26		simprr 784	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ (𝑠 ∈ 𝐵 ∧ (𝐹 “ 𝑠) ⊆ 𝑜)) → (𝐹 “ 𝑠) ⊆ 𝑜)
27		imaeq2 6049	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (𝐹 “ 𝑡) = (𝐹 “ 𝑠))
28	27	sseq1d 3970	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ((𝐹 “ 𝑡) ⊆ 𝑜 ↔ (𝐹 “ 𝑠) ⊆ 𝑜))
29	28	rspcev 3584	. . . . . . . 8 ⊢ ((𝑠 ∈ 𝐿 ∧ (𝐹 “ 𝑠) ⊆ 𝑜) → ∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜)
30	25, 26, 29	syl2anc 595	. . . . . . 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 3167	. . . . . 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 3188	. . 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 589	. 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))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089   ⊆ wss 3907   “ cima 5655  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  fBascfbas 21470  filGencfg 21471  TopOnctopon 23028  Filcfil 23963   fLimf cflf 24053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-map 8814  df-fbas 21479  df-fg 21480  df-top 23012  df-topon 23029  df-ntr 23138  df-nei 23216  df-fil 23964  df-fm 24056  df-flim 24057  df-flf 24058

This theorem is referenced by:  lmflf  24123  eltsms  24251