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Theorem limcfval 23617
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 23611 . . . 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 6190 . . . . 5 (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) → (TopOpen‘ℂfld) ∈ V)
4 simplrl 799 . . . . . . . . . 10 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑓 = 𝐹)
54dmeqd 5315 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → dom 𝑓 = dom 𝐹)
6 simpll1 1098 . . . . . . . . . 10 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝐹:𝐴⟶ℂ)
7 fdm 6038 . . . . . . . . . 10 (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴)
86, 7syl 17 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → dom 𝐹 = 𝐴)
95, 8eqtrd 2654 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → dom 𝑓 = 𝐴)
10 simplrr 800 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑥 = 𝐵)
1110sneqd 4180 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → {𝑥} = {𝐵})
129, 11uneq12d 3760 . . . . . . 7 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (dom 𝑓 ∪ {𝑥}) = (𝐴 ∪ {𝐵}))
1310eqeq2d 2630 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑧 = 𝑥𝑧 = 𝐵))
144fveq1d 6180 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑓𝑧) = (𝐹𝑧))
1513, 14ifbieq2d 4102 . . . . . . 7 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → if(𝑧 = 𝑥, 𝑦, (𝑓𝑧)) = if(𝑧 = 𝐵, 𝑦, (𝐹𝑧)))
1612, 15mpteq12dv 4724 . . . . . 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))))
17 simpr 477 . . . . . . . . . . 11 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑗 = (TopOpen‘ℂfld))
18 limcval.k . . . . . . . . . . 11 𝐾 = (TopOpen‘ℂfld)
1917, 18syl6eqr 2672 . . . . . . . . . 10 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑗 = 𝐾)
2019, 12oveq12d 6653 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑗t (dom 𝑓 ∪ {𝑥})) = (𝐾t (𝐴 ∪ {𝐵})))
21 limcval.j . . . . . . . . 9 𝐽 = (𝐾t (𝐴 ∪ {𝐵}))
2220, 21syl6eqr 2672 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑗t (dom 𝑓 ∪ {𝑥})) = 𝐽)
2322, 19oveq12d 6653 . . . . . . 7 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → ((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗) = (𝐽 CnP 𝐾))
2423, 10fveq12d 6184 . . . . . 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))
2516, 24eleq12d 2693 . . . . 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)))
263, 25sbcied 3466 . . . 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)))
2726abbidv 2739 . . 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)})
28 cnex 10002 . . . . 5 ℂ ∈ V
29 elpm2r 7860 . . . . 5 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
3028, 28, 29mpanl12 717 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
31303adant3 1079 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
32 simp3 1061 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
33 eqid 2620 . . . . . 6 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧)))
3421, 18, 33limcvallem 23616 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝑦 ∈ ℂ))
3534abssdv 3668 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ⊆ ℂ)
3628ssex 4793 . . . 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)
3735, 36syl 17 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ∈ V)
382, 27, 31, 32, 37ovmpt2d 6773 . 2 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐹 lim 𝐵) = {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)})
3938, 35eqsstrd 3631 . 2 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐹 lim 𝐵) ⊆ ℂ)
4038, 39jca 554 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 384  w3a 1036   = wceq 1481  wcel 1988  {cab 2606  Vcvv 3195  [wsbc 3429  cun 3565  wss 3567  ifcif 4077  {csn 4168  cmpt 4720  dom cdm 5104  wf 5872  cfv 5876  (class class class)co 6635  cmpt2 6637  pm cpm 7843  cc 9919  t crest 16062  TopOpenctopn 16063  fldccnfld 19727   CnP ccnp 21010   lim climc 23607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fi 8302  df-sup 8333  df-inf 8334  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-q 11774  df-rp 11818  df-xneg 11931  df-xadd 11932  df-xmul 11933  df-fz 12312  df-seq 12785  df-exp 12844  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-plusg 15935  df-mulr 15936  df-starv 15937  df-tset 15941  df-ple 15942  df-ds 15945  df-unif 15946  df-rest 16064  df-topn 16065  df-topgen 16085  df-psmet 19719  df-xmet 19720  df-met 19721  df-bl 19722  df-mopn 19723  df-cnfld 19728  df-top 20680  df-topon 20697  df-topsp 20718  df-bases 20731  df-cnp 21013  df-xms 22106  df-ms 22107  df-limc 23611
This theorem is referenced by:  ellimc  23618  limccl  23620
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