Theorem flftg 21800

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 21797	. 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 3142	. . . 4 ⊢ (∀𝑢 ∈ 𝐽 (𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ ∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
4		simpl1 1064	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
5		topontop 20718	. . . . . . . . 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	syl5eqelr 2706	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (topGen‘𝐵) ∈ Top)
8		tgclb 20774	. . . . . . 7 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
9	7, 8	sylibr 224	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐵 ∈ TopBases)
10		bastg 20770	. . . . . 6 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
11		eleq2 2690	. . . . . . . . 9 ⊢ (𝑢 = 𝑜 → (𝐴 ∈ 𝑢 ↔ 𝐴 ∈ 𝑜))
12		sseq2 3627	. . . . . . . . . 10 ⊢ (𝑢 = 𝑜 → ((𝐹 “ 𝑠) ⊆ 𝑢 ↔ (𝐹 “ 𝑠) ⊆ 𝑜))
13	12	rexbidv 3052	. . . . . . . . 9 ⊢ (𝑢 = 𝑜 → (∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢 ↔ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))
14	11, 13	imbi12d 334	. . . . . . . 8 ⊢ (𝑢 = 𝑜 → ((𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))
15	14	cbvralv 3171	. . . . . . 7 ⊢ (∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢) ↔ ∀𝑜 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))
16		ssralv 3666	. . . . . . 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 232	. . . . . 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 20769	. . . . . . . 8 ⊢ ((𝑢 ∈ (topGen‘𝐵) ∧ 𝐴 ∈ 𝑢) → ∃𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢))
20		r19.29 3072	. . . . . . . . . 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 473	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → 𝐴 ∈ 𝑜)
22		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢) → 𝑜 ⊆ 𝑢)
23		sstr2 3610	. . . . . . . . . . . . . . 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 3016	. . . . . . . . . . . . 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 446	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ 𝑢)) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢)
28	27	rexlimivw 3029	. . . . . . . . . 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 450	. . . . . . . 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 453	. . . . . 6 ⊢ ((∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) ∧ 𝑢 ∈ (topGen‘𝐵)) → (𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
33	32	ralrimiva 2966	. . . . 5 ⊢ (∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜) → ∀𝑢 ∈ (topGen‘𝐵)(𝐴 ∈ 𝑢 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑢))
34	18, 33	impbid1 215	. . . 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 272	. . 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 673	. 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 268	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  topGenctg 16098  Topctop 20698  TopOnctopon 20715  TopBasesctb 20749  Filcfil 21649   fLimf cflf 21739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-topgen 16104  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-bases 20750  df-ntr 20824  df-nei 20902  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744

This theorem is referenced by:  txflf  21810