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Theorem sslm 22810
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 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (β‡π‘‘β€˜πΎ) βŠ† (β‡π‘‘β€˜π½))
Proof of Theorem sslm
Dummy variables 𝑒 𝑓 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idd 24 . . . . 5 (𝐽 βŠ† 𝐾 β†’ (𝑓 ∈ (𝑋 ↑pm β„‚) β†’ 𝑓 ∈ (𝑋 ↑pm β„‚)))
2 idd 24 . . . . 5 (𝐽 βŠ† 𝐾 β†’ (π‘₯ ∈ 𝑋 β†’ π‘₯ ∈ 𝑋))
3 ssralv 4050 . . . . 5 (𝐽 βŠ† 𝐾 β†’ (βˆ€π‘’ ∈ 𝐾 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’) β†’ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
41, 2, 33anim123d 1443 . . . 4 (𝐽 βŠ† 𝐾 β†’ ((𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐾 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’)) β†’ (𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))))
54ssopab2dv 5551 . . 3 (𝐽 βŠ† 𝐾 β†’ {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐾 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))} βŠ† {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))})
653ad2ant3 1135 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐾 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))} βŠ† {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))})
7 lmfval 22743 . . 3 (𝐾 ∈ (TopOnβ€˜π‘‹) β†’ (β‡π‘‘β€˜πΎ) = {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐾 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))})
873ad2ant2 1134 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (β‡π‘‘β€˜πΎ) = {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐾 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))})
9 lmfval 22743 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (β‡π‘‘β€˜π½) = {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))})
1093ad2ant1 1133 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (β‡π‘‘β€˜π½) = {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))})
116, 8, 103sstr4d 4029 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (β‡π‘‘β€˜πΎ) βŠ† (β‡π‘‘β€˜π½))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3948  {copab 5210  ran crn 5677   β†Ύ cres 5678  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ↑pm cpm 8823  β„‚cc 11110  β„€β‰₯cuz 12824  TopOnctopon 22419  β‡π‘‘clm 22737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-top 22403  df-topon 22420  df-lm 22740
This theorem is referenced by: (None)
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