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Theorem sslm 21481
Description: A finer topology has fewer convergent sequences (but the sequences that do converge, converge to the same value). (Contributed by Mario Carneiro, 15-Sep-2015.)
Assertion
Ref Expression
sslm ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (⇝𝑡𝐾) ⊆ (⇝𝑡𝐽))

Proof of Theorem sslm
Dummy variables 𝑢 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idd 24 . . . . 5 (𝐽𝐾 → (𝑓 ∈ (𝑋pm ℂ) → 𝑓 ∈ (𝑋pm ℂ)))
2 idd 24 . . . . 5 (𝐽𝐾 → (𝑥𝑋𝑥𝑋))
3 ssralv 3891 . . . . 5 (𝐽𝐾 → (∀𝑢𝐾 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢) → ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)))
41, 2, 33anim123d 1571 . . . 4 (𝐽𝐾 → ((𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐾 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)) → (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))))
54ssopab2dv 5232 . . 3 (𝐽𝐾 → {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐾 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))} ⊆ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
653ad2ant3 1169 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐾 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))} ⊆ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
7 lmfval 21414 . . 3 (𝐾 ∈ (TopOn‘𝑋) → (⇝𝑡𝐾) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐾 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
873ad2ant2 1168 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (⇝𝑡𝐾) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐾 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
9 lmfval 21414 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
1093ad2ant1 1167 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
116, 8, 103sstr4d 3873 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (⇝𝑡𝐾) ⊆ (⇝𝑡𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1111   = wceq 1656  wcel 2164  wral 3117  wrex 3118  wss 3798  {copab 4937  ran crn 5347  cres 5348  wf 6123  cfv 6127  (class class class)co 6910  pm cpm 8128  cc 10257  cuz 11975  TopOnctopon 21092  𝑡clm 21408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-ov 6913  df-top 21076  df-topon 21093  df-lm 21411
This theorem is referenced by: (None)
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