Theorem flffbas 22605

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 22488	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
3	1, 2	eqeltrid 2919	. . 3 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
4		isflf 22603	. . 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 1160	. 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 2906	. . . . . . . 8 ⊢ (𝑡 ∈ 𝐿 ↔ 𝑡 ∈ (𝑌filGen𝐵))
7		elfg 22481	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑡 ∈ (𝑌filGen𝐵) ↔ (𝑡 ⊆ 𝑌 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
8	7	3ad2ant2 1130	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑡 ∈ (𝑌filGen𝐵) ↔ (𝑡 ⊆ 𝑌 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑡)))
9		sstr2 3976	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 “ 𝑠) ⊆ (𝐹 “ 𝑡) → ((𝐹 “ 𝑡) ⊆ 𝑜 → (𝐹 “ 𝑠) ⊆ 𝑜))
10		imass2 5967	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ⊆ 𝑡 → (𝐹 “ 𝑠) ⊆ (𝐹 “ 𝑡))
11	9, 10	syl11 33	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 “ 𝑡) ⊆ 𝑜 → (𝑠 ⊆ 𝑡 → (𝐹 “ 𝑠) ⊆ 𝑜))
12	11	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐹 “ 𝑡) ⊆ 𝑜) → (𝑠 ⊆ 𝑡 → (𝐹 “ 𝑠) ⊆ 𝑜))
13	12	reximdv 3275	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 86	. . . . . . . . . . 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 493	. . . . . . . . . 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 242	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑡 ∈ (𝑌filGen𝐵) → ((𝐹 “ 𝑡) ⊆ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜)))
18	17	adantr 483	. . . . . . . 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	syl5bi 244	. . . . . . 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 3285	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜))
21		ssfg 22482	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
22	21, 1	sseqtrrdi 4020	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ 𝐿)
23	22	sselda 3969	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ 𝐿)
24	23	3ad2antl2 1182	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ 𝐿)
25	24	ad2ant2r 745	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ (𝑠 ∈ 𝐵 ∧ (𝐹 “ 𝑠) ⊆ 𝑜)) → 𝑠 ∈ 𝐿)
26		simprr 771	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ (𝑠 ∈ 𝐵 ∧ (𝐹 “ 𝑠) ⊆ 𝑜)) → (𝐹 “ 𝑠) ⊆ 𝑜)
27		imaeq2 5927	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (𝐹 “ 𝑡) = (𝐹 “ 𝑠))
28	27	sseq1d 4000	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ((𝐹 “ 𝑡) ⊆ 𝑜 ↔ (𝐹 “ 𝑠) ⊆ 𝑜))
29	28	rspcev 3625	. . . . . . . 8 ⊢ ((𝑠 ∈ 𝐿 ∧ (𝐹 “ 𝑠) ⊆ 𝑜) → ∃𝑡 ∈ 𝐿 (𝐹 “ 𝑡) ⊆ 𝑜)
30	25, 26, 29	syl2anc 586	. . . . . . 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 3287	. . . . . 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 214	. . . . 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 3199	. . 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 581	. 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 281	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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ⊆ wss 3938   “ cima 5560  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  fBascfbas 20535  filGencfg 20536  TopOnctopon 21520  Filcfil 22455   fLimf cflf 22545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410  df-fbas 20544  df-fg 20545  df-top 21504  df-topon 21521  df-ntr 21630  df-nei 21708  df-fil 22456  df-fm 22548  df-flim 22549  df-flf 22550

This theorem is referenced by:  lmflf  22615  eltsms  22743