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Theorem flimcf 21958
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)) → (𝐽𝐾 ↔ ∀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.
StepHypRef Expression
1 simplll 815 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝐽 ∈ (TopOn‘𝑋))
2 simprl 811 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝑓 ∈ (Fil‘𝑋))
3 simplr 809 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝐽𝐾)
4 flimss1 21949 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓))
51, 2, 3, 4syl3anc 1463 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → (𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓))
6 simprr 813 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝑥 ∈ (𝐾 fLim 𝑓))
75, 6sseldd 3733 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝑥 ∈ (𝐽 fLim 𝑓))
87expr 644 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐾 fLim 𝑓) → 𝑥 ∈ (𝐽 fLim 𝑓)))
98ssrdv 3738 . . 3 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (Fil‘𝑋)) → (𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓))
109ralrimiva 3092 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽𝐾) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓))
11 simpllr 817 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → 𝐾 ∈ (TopOn‘𝑋))
12 simplll 815 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
13 simprl 811 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥𝐽)
14 toponss 20904 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
1512, 13, 14syl2anc 696 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥𝑋)
16 simprr 813 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦𝑥)
1715, 16sseldd 3733 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦𝑋)
1817snssd 4473 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → {𝑦} ⊆ 𝑋)
19 snnzg 4439 . . . . . . . . . . . . 13 (𝑦𝑋 → {𝑦} ≠ ∅)
2017, 19syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → {𝑦} ≠ ∅)
21 neifil 21856 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOn‘𝑋) ∧ {𝑦} ⊆ 𝑋 ∧ {𝑦} ≠ ∅) → ((nei‘𝐾)‘{𝑦}) ∈ (Fil‘𝑋))
2211, 18, 20, 21syl3anc 1463 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → ((nei‘𝐾)‘{𝑦}) ∈ (Fil‘𝑋))
23 simplr 809 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓))
24 oveq2 6809 . . . . . . . . . . . . 13 (𝑓 = ((nei‘𝐾)‘{𝑦}) → (𝐾 fLim 𝑓) = (𝐾 fLim ((nei‘𝐾)‘{𝑦})))
25 oveq2 6809 . . . . . . . . . . . . 13 (𝑓 = ((nei‘𝐾)‘{𝑦}) → (𝐽 fLim 𝑓) = (𝐽 fLim ((nei‘𝐾)‘{𝑦})))
2624, 25sseq12d 3763 . . . . . . . . . . . 12 (𝑓 = ((nei‘𝐾)‘{𝑦}) → ((𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓) ↔ (𝐾 fLim ((nei‘𝐾)‘{𝑦})) ⊆ (𝐽 fLim ((nei‘𝐾)‘{𝑦}))))
2726rspcv 3433 . . . . . . . . . . 11 (((nei‘𝐾)‘{𝑦}) ∈ (Fil‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓) → (𝐾 fLim ((nei‘𝐾)‘{𝑦})) ⊆ (𝐽 fLim ((nei‘𝐾)‘{𝑦}))))
2822, 23, 27sylc 65 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → (𝐾 fLim ((nei‘𝐾)‘{𝑦})) ⊆ (𝐽 fLim ((nei‘𝐾)‘{𝑦})))
29 neiflim 21950 . . . . . . . . . . 11 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑦𝑋) → 𝑦 ∈ (𝐾 fLim ((nei‘𝐾)‘{𝑦})))
3011, 17, 29syl2anc 696 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 ∈ (𝐾 fLim ((nei‘𝐾)‘{𝑦})))
3128, 30sseldd 3733 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 ∈ (𝐽 fLim ((nei‘𝐾)‘{𝑦})))
32 flimneiss 21942 . . . . . . . . 9 (𝑦 ∈ (𝐽 fLim ((nei‘𝐾)‘{𝑦})) → ((nei‘𝐽)‘{𝑦}) ⊆ ((nei‘𝐾)‘{𝑦}))
3331, 32syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → ((nei‘𝐽)‘{𝑦}) ⊆ ((nei‘𝐾)‘{𝑦}))
34 topontop 20891 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3512, 34syl 17 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Top)
36 opnneip 21096 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑦𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝑦}))
3735, 13, 16, 36syl3anc 1463 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝑦}))
3833, 37sseldd 3733 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥 ∈ ((nei‘𝐾)‘{𝑦}))
3938anassrs 683 . . . . . 6 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥𝐽) ∧ 𝑦𝑥) → 𝑥 ∈ ((nei‘𝐾)‘{𝑦}))
4039ralrimiva 3092 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥𝐽) → ∀𝑦𝑥 𝑥 ∈ ((nei‘𝐾)‘{𝑦}))
41 simpllr 817 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥𝐽) → 𝐾 ∈ (TopOn‘𝑋))
42 topontop 20891 . . . . . 6 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
43 opnnei 21097 . . . . . 6 (𝐾 ∈ Top → (𝑥𝐾 ↔ ∀𝑦𝑥 𝑥 ∈ ((nei‘𝐾)‘{𝑦})))
4441, 42, 433syl 18 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥𝐽) → (𝑥𝐾 ↔ ∀𝑦𝑥 𝑥 ∈ ((nei‘𝐾)‘{𝑦})))
4540, 44mpbird 247 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥𝐽) → 𝑥𝐾)
4645ex 449 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) → (𝑥𝐽𝑥𝐾))
4746ssrdv 3738 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) → 𝐽𝐾)
4810, 47impbida 913 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1620  wcel 2127  wne 2920  wral 3038  wss 3703  c0 4046  {csn 4309  cfv 6037  (class class class)co 6801  Topctop 20871  TopOnctopon 20888  neicnei 21074  Filcfil 21821   fLim cflim 21910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-fbas 19916  df-top 20872  df-topon 20889  df-ntr 20997  df-nei 21075  df-fil 21822  df-flim 21915
This theorem is referenced by: (None)
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