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Theorem lmfval 22956
Description: The relation "sequence 𝑓 converges to point 𝑦 " in a metric space. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
lmfval (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‘’))})
Distinct variable groups:   π‘₯,𝑓,𝑦,𝑋   𝑒,𝑓,𝐽,π‘₯,𝑦
Allowed substitution hint:   𝑋(𝑒)
Proof of Theorem lmfval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 df-lm 22953 . 2 ⇝𝑑 = (𝑗 ∈ Top ↦ {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (βˆͺ 𝑗 ↑pm β„‚) ∧ π‘₯ ∈ βˆͺ 𝑗 ∧ βˆ€π‘’ ∈ 𝑗 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))})
2 simpr 483 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ 𝑗 = 𝐽)
32unieqd 4921 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
4 toponuni 22636 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
54adantr 479 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ 𝑋 = βˆͺ 𝐽)
63, 5eqtr4d 2773 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ βˆͺ 𝑗 = 𝑋)
76oveq1d 7426 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ (βˆͺ 𝑗 ↑pm β„‚) = (𝑋 ↑pm β„‚))
87eleq2d 2817 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ (𝑓 ∈ (βˆͺ 𝑗 ↑pm β„‚) ↔ 𝑓 ∈ (𝑋 ↑pm β„‚)))
96eleq2d 2817 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ (π‘₯ ∈ βˆͺ 𝑗 ↔ π‘₯ ∈ 𝑋))
102raleqdv 3323 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ (βˆ€π‘’ ∈ 𝑗 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’) ↔ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
118, 9, 103anbi123d 1434 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ ((𝑓 ∈ (βˆͺ 𝑗 ↑pm β„‚) ∧ π‘₯ ∈ βˆͺ 𝑗 ∧ βˆ€π‘’ ∈ 𝑗 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’)) ↔ (𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))))
1211opabbidv 5213 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (βˆͺ 𝑗 ↑pm β„‚) ∧ π‘₯ ∈ βˆͺ 𝑗 ∧ βˆ€π‘’ ∈ 𝑗 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))} = {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‘’))})
13 topontop 22635 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
14 df-3an 1087 . . . . 5 ((𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’)) ↔ ((𝑓 ∈ (𝑋 ↑pm β„‚) ∧ π‘₯ ∈ 𝑋) ∧ βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
1514opabbii 5214 . . . 4 {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‘’))} = {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‘’))}
16 opabssxp 5767 . . . 4 {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‘’))} βŠ† ((𝑋 ↑pm β„‚) Γ— 𝑋)
1715, 16eqsstri 4015 . . 3 {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‘’))} βŠ† ((𝑋 ↑pm β„‚) Γ— 𝑋)
18 ovex 7444 . . . 4 (𝑋 ↑pm β„‚) ∈ V
19 toponmax 22648 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
20 xpexg 7739 . . . 4 (((𝑋 ↑pm β„‚) ∈ V ∧ 𝑋 ∈ 𝐽) β†’ ((𝑋 ↑pm β„‚) Γ— 𝑋) ∈ V)
2118, 19, 20sylancr 585 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ((𝑋 ↑pm β„‚) Γ— 𝑋) ∈ V)
22 ssexg 5322 . . 3 (({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‘’))} βŠ† ((𝑋 ↑pm β„‚) Γ— 𝑋) ∧ ((𝑋 ↑pm β„‚) Γ— 𝑋) ∈ V) β†’ {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‘’))} ∈ V)
2317, 21, 22sylancr 585 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ {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‘’))} ∈ V)
241, 12, 13, 23fvmptd2 7005 1 (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‘’))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βŠ† wss 3947  βˆͺ cuni 4907  {copab 5209   Γ— cxp 5673  ran crn 5676   β†Ύ cres 5677  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ↑pm cpm 8823  β„‚cc 11110  β„€β‰₯cuz 12826  Topctop 22615  TopOnctopon 22632  β‡π‘‘clm 22950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-top 22616  df-topon 22633  df-lm 22953
This theorem is referenced by:  lmbr  22982  sslm  23023
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