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Theorem lmfval 13695
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 13693 . . 3 ⇝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‘’))})
21a1i 9 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ⇝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‘’))}))
3 simpr 110 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ 𝑗 = 𝐽)
43unieqd 3821 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
5 toponuni 13518 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
65adantr 276 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ 𝑋 = βˆͺ 𝐽)
74, 6eqtr4d 2213 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ βˆͺ 𝑗 = 𝑋)
87oveq1d 5890 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ (βˆͺ 𝑗 ↑pm β„‚) = (𝑋 ↑pm β„‚))
98eleq2d 2247 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ (𝑓 ∈ (βˆͺ 𝑗 ↑pm β„‚) ↔ 𝑓 ∈ (𝑋 ↑pm β„‚)))
107eleq2d 2247 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑗 = 𝐽) β†’ (π‘₯ ∈ βˆͺ 𝑗 ↔ π‘₯ ∈ 𝑋))
113raleqdv 2679 . . . 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‘’)))
129, 10, 113anbi123d 1312 . . 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‘’))))
1312opabbidv 4070 . 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‘’))})
14 topontop 13517 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
15 df-3an 980 . . . . 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‘’)))
1615opabbii 4071 . . . 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‘’))}
17 opabssxp 4701 . . . 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)
1816, 17eqsstri 3188 . . 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)
19 fnpm 6656 . . . . 5 ↑pm Fn (V Γ— V)
20 toponmax 13528 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
2120elexd 2751 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ V)
22 cnex 7935 . . . . . 6 β„‚ ∈ V
2322a1i 9 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ β„‚ ∈ V)
24 fnovex 5908 . . . . 5 (( ↑pm Fn (V Γ— V) ∧ 𝑋 ∈ V ∧ β„‚ ∈ V) β†’ (𝑋 ↑pm β„‚) ∈ V)
2519, 21, 23, 24mp3an2i 1342 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝑋 ↑pm β„‚) ∈ V)
26 xpexg 4741 . . . 4 (((𝑋 ↑pm β„‚) ∈ V ∧ 𝑋 ∈ 𝐽) β†’ ((𝑋 ↑pm β„‚) Γ— 𝑋) ∈ V)
2725, 20, 26syl2anc 411 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ((𝑋 ↑pm β„‚) Γ— 𝑋) ∈ V)
28 ssexg 4143 . . 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)
2918, 27, 28sylancr 414 . 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)
302, 13, 14, 29fvmptd 5598 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 set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  βˆƒwrex 2456  Vcvv 2738   βŠ† wss 3130  βˆͺ cuni 3810  {copab 4064   ↦ cmpt 4065   Γ— cxp 4625  ran crn 4628   β†Ύ cres 4629   Fn wfn 5212  βŸΆwf 5213  β€˜cfv 5217  (class class class)co 5875   ↑pm cpm 6649  β„‚cc 7809  β„€β‰₯cuz 9528  Topctop 13500  TopOnctopon 13513  β‡π‘‘clm 13690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-cnex 7902
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-pm 6651  df-top 13501  df-topon 13514  df-lm 13693
This theorem is referenced by:  lmreltop  13696  lmbr  13716  sslm  13750
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