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Theorem lmbr 22995
Description: Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition 𝐹 βŠ† (β„‚ Γ— 𝑋) allows to use objects more general than sequences when convenient; see the comment in df-lm 22966. (Contributed by Mario Carneiro, 14-Nov-2013.)
Hypothesis
Ref Expression
lmbr.2 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
Assertion
Ref Expression
lmbr (πœ‘ β†’ (𝐹(β‡π‘‘β€˜π½)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’))))
Distinct variable groups:   𝑦,𝑒,𝐹   𝑒,𝐽,𝑦   πœ‘,𝑒   𝑒,𝑃   𝑒,𝑋,𝑦
Allowed substitution hints:   πœ‘(𝑦)   𝑃(𝑦)
Proof of Theorem lmbr
Dummy variables 𝑓 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmbr.2 . . . 4 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 lmfval 22969 . . . 4 (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‘’))})
31, 2syl 17 . . 3 (πœ‘ 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‘’))})
43breqd 5159 . 2 (πœ‘ β†’ (𝐹(β‡π‘‘β€˜π½)𝑃 ↔ 𝐹{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‘’))}𝑃))
5 reseq1 5975 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (𝑓 β†Ύ 𝑦) = (𝐹 β†Ύ 𝑦))
65feq1d 6702 . . . . . . . 8 (𝑓 = 𝐹 β†’ ((𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’ ↔ (𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’))
76rexbidv 3177 . . . . . . 7 (𝑓 = 𝐹 β†’ (βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’ ↔ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’))
87imbi2d 340 . . . . . 6 (𝑓 = 𝐹 β†’ ((π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’) ↔ (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
98ralbidv 3176 . . . . 5 (𝑓 = 𝐹 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‘’)))
10 eleq1 2820 . . . . . . 7 (π‘₯ = 𝑃 β†’ (π‘₯ ∈ 𝑒 ↔ 𝑃 ∈ 𝑒))
1110imbi1d 341 . . . . . 6 (π‘₯ = 𝑃 β†’ ((π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’) ↔ (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
1211ralbidv 3176 . . . . 5 (π‘₯ = 𝑃 β†’ (βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’) ↔ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
139, 12sylan9bb 509 . . . 4 ((𝑓 = 𝐹 ∧ π‘₯ = 𝑃) β†’ (βˆ€π‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’) ↔ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
14 df-3an 1088 . . . . 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 5215 . . . 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‘’))}
1613, 15brab2a 5769 . . 3 (𝐹{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‘’))}𝑃 ↔ ((𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋) ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
17 df-3an 1088 . . 3 ((𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)) ↔ ((𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋) ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
1816, 17bitr4i 278 . 2 (𝐹{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‘’))}𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
194, 18bitrdi 287 1 (πœ‘ β†’ (𝐹(β‡π‘‘β€˜π½)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069   class class class wbr 5148  {copab 5210  ran crn 5677   β†Ύ cres 5678  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412   ↑pm cpm 8827  β„‚cc 11114  β„€β‰₯cuz 12829  TopOnctopon 22645  β‡π‘‘clm 22963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-top 22629  df-topon 22646  df-lm 22966
This theorem is referenced by:  lmbr2  22996  lmfpm  23032  lmcl  23034  lmff  23038  lmmbr  25019
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