Theorem flimcf 24313

Description: Fineness is properly characterized by the property that every limit point of a filter in the finer topology is a limit point in the coarser topology. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
flimcf	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)))

Distinct variable groups:   𝑓,𝐽   𝑓,𝐾   𝑓,𝑋

Proof of Theorem flimcf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplll 787	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝐽 ∈ (TopOn‘𝑋))
2		simprl 783	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝑓 ∈ (Fil‘𝑋))
3		simplr 781	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝐽 ⊆ 𝐾)
4		flimss1 24304	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓))
5	1, 2, 3, 4	syl3anc 1398	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → (𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓))
6		simprr 785	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝑥 ∈ (𝐾 fLim 𝑓))
7	5, 6	sseldd 3913	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝑥 ∈ (𝐽 fLim 𝑓))
8	7	expr 462	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐾 fLim 𝑓) → 𝑥 ∈ (𝐽 fLim 𝑓)))
9	8	ssrdv 3918	. . 3 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (Fil‘𝑋)) → (𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓))
10	9	ralrimiva 3157	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓))
11		oveq2 7435	. . . . . . . . . . . 12 ⊢ (𝑓 = ((nei‘𝐾)‘{𝑦}) → (𝐾 fLim 𝑓) = (𝐾 fLim ((nei‘𝐾)‘{𝑦})))
12		oveq2 7435	. . . . . . . . . . . 12 ⊢ (𝑓 = ((nei‘𝐾)‘{𝑦}) → (𝐽 fLim 𝑓) = (𝐽 fLim ((nei‘𝐾)‘{𝑦})))
13	11, 12	sseq12d 3945	. . . . . . . . . . 11 ⊢ (𝑓 = ((nei‘𝐾)‘{𝑦}) → ((𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓) ↔ (𝐾 fLim ((nei‘𝐾)‘{𝑦})) ⊆ (𝐽 fLim ((nei‘𝐾)‘{𝑦}))))
14		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓))
15		simpllr 788	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐾 ∈ (TopOn‘𝑋))
16		simplll 787	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
17		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐽)
18		toponss 23257	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
19	16, 17, 18	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ⊆ 𝑋)
20		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
21	19, 20	sseldd 3913	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑋)
22	21	snssd 4746	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑦} ⊆ 𝑋)
23	20	snn0d 4735	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑦} ≠ ∅)
24		neifil 24211	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ {𝑦} ⊆ 𝑋 ∧ {𝑦} ≠ ∅) → ((nei‘𝐾)‘{𝑦}) ∈ (Fil‘𝑋))
25	15, 22, 23, 24	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ((nei‘𝐾)‘{𝑦}) ∈ (Fil‘𝑋))
26	13, 14, 25	rspcdva 3580	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (𝐾 fLim ((nei‘𝐾)‘{𝑦})) ⊆ (𝐽 fLim ((nei‘𝐾)‘{𝑦})))
27		neiflim 24305	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ (𝐾 fLim ((nei‘𝐾)‘{𝑦})))
28	15, 21, 27	syl2anc 596	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ (𝐾 fLim ((nei‘𝐾)‘{𝑦})))
29	26, 28	sseldd 3913	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ (𝐽 fLim ((nei‘𝐾)‘{𝑦})))
30		flimneiss 24297	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐽 fLim ((nei‘𝐾)‘{𝑦})) → ((nei‘𝐽)‘{𝑦}) ⊆ ((nei‘𝐾)‘{𝑦}))
31	29, 30	syl 18	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ((nei‘𝐽)‘{𝑦}) ⊆ ((nei‘𝐾)‘{𝑦}))
32		topontop 23243	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
33	16, 32	syl 18	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Top)
34		opnneip 23449	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝑦}))
35	33, 17, 20, 34	syl3anc 1398	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝑦}))
36	31, 35	sseldd 3913	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ ((nei‘𝐾)‘{𝑦}))
37	36	anassrs 473	. . . . . 6 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ ((nei‘𝐾)‘{𝑦}))
38	37	ralrimiva 3157	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥 ∈ 𝐽) → ∀𝑦 ∈ 𝑥 𝑥 ∈ ((nei‘𝐾)‘{𝑦}))
39		simpllr 788	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥 ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑋))
40		topontop 23243	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
41		opnnei 23450	. . . . . 6 ⊢ (𝐾 ∈ Top → (𝑥 ∈ 𝐾 ↔ ∀𝑦 ∈ 𝑥 𝑥 ∈ ((nei‘𝐾)‘{𝑦})))
42	39, 40, 41	3syl 19	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∈ 𝐾 ↔ ∀𝑦 ∈ 𝑥 𝑥 ∈ ((nei‘𝐾)‘{𝑦})))
43	38, 42	mpbird 260	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐾)
44	43	ex 418	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) → (𝑥 ∈ 𝐽 → 𝑥 ∈ 𝐾))
45	44	ssrdv 3918	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) → 𝐽 ⊆ 𝐾)
46	10, 45	impbida 813	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146   ≠ wne 2958  ∀wral 3079   ⊆ wss 3899  ∅c0 4280  {csn 4587  ‘cfv 6546  (class class class)co 7427  Topctop 23223  TopOnctopon 23240  neicnei 23427  Filcfil 24176   fLim cflim 24265

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-fbas 21687  df-top 23224  df-topon 23241  df-ntr 23350  df-nei 23428  df-fil 24177  df-flim 24270

This theorem is used by: (None)