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Theorem limcfval 24475
Description: Value and set bounds on the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
limcval.j 𝐽 = (𝐾t (𝐴 ∪ {𝐵}))
limcval.k 𝐾 = (TopOpen‘ℂfld)
Assertion
Ref Expression
limcfval ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 lim 𝐵) = {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ∧ (𝐹 lim 𝐵) ⊆ ℂ))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑦,𝐹,𝑧   𝑦,𝐾,𝑧   𝑦,𝐽
Allowed substitution hint:   𝐽(𝑧)

Proof of Theorem limcfval
Dummy variables 𝑓 𝑗 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-limc 24469 . . . 4 lim = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦[(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)})
21a1i 11 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → lim = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦[(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)}))
3 fvexd 6660 . . . . 5 (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) → (TopOpen‘ℂfld) ∈ V)
4 simplrl 776 . . . . . . . . . 10 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑓 = 𝐹)
54dmeqd 5738 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → dom 𝑓 = dom 𝐹)
6 simpll1 1209 . . . . . . . . . 10 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝐹:𝐴⟶ℂ)
76fdmd 6497 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → dom 𝐹 = 𝐴)
85, 7eqtrd 2833 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → dom 𝑓 = 𝐴)
9 simplrr 777 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑥 = 𝐵)
109sneqd 4537 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → {𝑥} = {𝐵})
118, 10uneq12d 4091 . . . . . . 7 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (dom 𝑓 ∪ {𝑥}) = (𝐴 ∪ {𝐵}))
129eqeq2d 2809 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑧 = 𝑥𝑧 = 𝐵))
134fveq1d 6647 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑓𝑧) = (𝐹𝑧))
1412, 13ifbieq2d 4450 . . . . . . 7 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → if(𝑧 = 𝑥, 𝑦, (𝑓𝑧)) = if(𝑧 = 𝐵, 𝑦, (𝐹𝑧)))
1511, 14mpteq12dv 5115 . . . . . 6 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))))
16 simpr 488 . . . . . . . . . . 11 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑗 = (TopOpen‘ℂfld))
17 limcval.k . . . . . . . . . . 11 𝐾 = (TopOpen‘ℂfld)
1816, 17eqtr4di 2851 . . . . . . . . . 10 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑗 = 𝐾)
1918, 11oveq12d 7153 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑗t (dom 𝑓 ∪ {𝑥})) = (𝐾t (𝐴 ∪ {𝐵})))
20 limcval.j . . . . . . . . 9 𝐽 = (𝐾t (𝐴 ∪ {𝐵}))
2119, 20eqtr4di 2851 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑗t (dom 𝑓 ∪ {𝑥})) = 𝐽)
2221, 18oveq12d 7153 . . . . . . 7 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → ((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗) = (𝐽 CnP 𝐾))
2322, 9fveq12d 6652 . . . . . 6 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥) = ((𝐽 CnP 𝐾)‘𝐵))
2415, 23eleq12d 2884 . . . . 5 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → ((𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)))
253, 24sbcied 3762 . . . 4 (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) → ([(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)))
2625abbidv 2862 . . 3 (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) → {𝑦[(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)} = {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)})
27 cnex 10607 . . . . 5 ℂ ∈ V
28 elpm2r 8407 . . . . 5 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
2927, 27, 28mpanl12 701 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
30293adant3 1129 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
31 simp3 1135 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
32 eqid 2798 . . . . . 6 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧)))
3320, 17, 32limcvallem 24474 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝑦 ∈ ℂ))
3433abssdv 3996 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ⊆ ℂ)
3527ssex 5189 . . . 4 ({𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ⊆ ℂ → {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ∈ V)
3634, 35syl 17 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ∈ V)
372, 26, 30, 31, 36ovmpod 7281 . 2 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐹 lim 𝐵) = {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)})
3837, 34eqsstrd 3953 . 2 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐹 lim 𝐵) ⊆ ℂ)
3937, 38jca 515 1 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 lim 𝐵) = {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ∧ (𝐹 lim 𝐵) ⊆ ℂ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  {cab 2776  Vcvv 3441  [wsbc 3720  cun 3879  wss 3881  ifcif 4425  {csn 4525  cmpt 5110  dom cdm 5519  wf 6320  cfv 6324  (class class class)co 7135  cmpo 7137  pm cpm 8390  cc 10524  t crest 16686  TopOpenctopn 16687  fldccnfld 20091   CnP ccnp 21830   lim climc 24465
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-sup 8890  df-inf 8891  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-fz 12886  df-seq 13365  df-exp 13426  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-plusg 16570  df-mulr 16571  df-starv 16572  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-rest 16688  df-topn 16689  df-topgen 16709  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-cnfld 20092  df-top 21499  df-topon 21516  df-topsp 21538  df-bases 21551  df-cnp 21833  df-xms 22927  df-ms 22928  df-limc 24469
This theorem is referenced by:  ellimc  24476  limccl  24478
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