Theorem flftg 22607

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 22604	. 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 3416	. . . 4 ⊢ (∀𝑢 ∈ 𝐽 (𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ ∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
4		simpl1 1187	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
5		topontop 21524	. . . . . . . . 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 2921	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (topGen‘𝐵) ∈ Top)
8		tgclb 21581	. . . . . . 7 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
9	7, 8	sylibr 236	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐵 ∈ TopBases)
10		bastg 21577	. . . . . 6 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
11		eleq2w 2899	. . . . . . . . 9 ⊢ (𝑢 = 𝑜 → (𝐴 ∈ 𝑢 ↔ 𝐴 ∈ 𝑜))
12		sseq2 3996	. . . . . . . . . 10 ⊢ (𝑢 = 𝑜 → ((𝐹 “ 𝑠) ⊆ 𝑢 ↔ (𝐹 “ 𝑠) ⊆ 𝑜))
13	12	rexbidv 3300	. . . . . . . . 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 3452	. . . . . . 7 ⊢ (∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ ∀𝑜 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))
16		ssralv 4036	. . . . . . 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	syl5bi 244	. . . . . 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 21576	. . . . . . . 8 ⊢ ((𝑢 ∈ (topGen‘𝐵) ∧ 𝐴 ∈ 𝑢) → ∃𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢))
20		r19.29 3257	. . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → 𝐴 ∈ 𝑜)
22		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → 𝑜 ⊆ 𝑢)
23		sstr2 3977	. . . . . . . . . . . . . . 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 3276	. . . . . . . . . . . . 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 410	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢)) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢)
28	27	rexlimivw 3285	. . . . . . . . . 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 415	. . . . . . . 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 455	. . . . . 6 ⊢ ((∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ 𝑢 ∈ (topGen‘𝐵)) → (𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
33	32	ralrimiva 3185	. . . . 5 ⊢ (∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) → ∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
34	18, 33	impbid1 227	. . . 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	syl5bb 285	. . 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 581	. 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 281	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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ∃wrex 3142   ⊆ wss 3939   “ cima 5561  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  topGenctg 16714  Topctop 21504  TopOnctopon 21521  TopBasesctb 21556  Filcfil 22456   fLimf cflf 22546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-map 8411  df-topgen 16720  df-fbas 20545  df-fg 20546  df-top 21505  df-topon 21522  df-bases 21557  df-ntr 21631  df-nei 21709  df-fil 22457  df-fm 22549  df-flim 22550  df-flf 22551

This theorem is referenced by:  txflf  22617