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Theorem limcfval 23356
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 23350 . . . 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 fvex 6095 . . . . . 6 (TopOpen‘ℂfld) ∈ V
43a1i 11 . . . . 5 (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) → (TopOpen‘ℂfld) ∈ V)
5 simplrl 795 . . . . . . . . . 10 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑓 = 𝐹)
65dmeqd 5232 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → dom 𝑓 = dom 𝐹)
7 simpll1 1092 . . . . . . . . . 10 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝐹:𝐴⟶ℂ)
8 fdm 5947 . . . . . . . . . 10 (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴)
97, 8syl 17 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → dom 𝐹 = 𝐴)
106, 9eqtrd 2640 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → dom 𝑓 = 𝐴)
11 simplrr 796 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑥 = 𝐵)
1211sneqd 4133 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → {𝑥} = {𝐵})
1310, 12uneq12d 3726 . . . . . . 7 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (dom 𝑓 ∪ {𝑥}) = (𝐴 ∪ {𝐵}))
1411eqeq2d 2616 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑧 = 𝑥𝑧 = 𝐵))
155fveq1d 6087 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑓𝑧) = (𝐹𝑧))
1614, 15ifbieq2d 4057 . . . . . . 7 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → if(𝑧 = 𝑥, 𝑦, (𝑓𝑧)) = if(𝑧 = 𝐵, 𝑦, (𝐹𝑧)))
1713, 16mpteq12dv 4654 . . . . . 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))))
18 simpr 475 . . . . . . . . . . 11 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑗 = (TopOpen‘ℂfld))
19 limcval.k . . . . . . . . . . 11 𝐾 = (TopOpen‘ℂfld)
2018, 19syl6eqr 2658 . . . . . . . . . 10 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑗 = 𝐾)
2120, 13oveq12d 6542 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑗t (dom 𝑓 ∪ {𝑥})) = (𝐾t (𝐴 ∪ {𝐵})))
22 limcval.j . . . . . . . . 9 𝐽 = (𝐾t (𝐴 ∪ {𝐵}))
2321, 22syl6eqr 2658 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑗t (dom 𝑓 ∪ {𝑥})) = 𝐽)
2423, 20oveq12d 6542 . . . . . . 7 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → ((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗) = (𝐽 CnP 𝐾))
2524, 11fveq12d 6091 . . . . . 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))
2617, 25eleq12d 2678 . . . . 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)))
274, 26sbcied 3435 . . . 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)))
2827abbidv 2724 . . 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)})
29 cnex 9870 . . . . 5 ℂ ∈ V
30 elpm2r 7735 . . . . 5 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
3129, 29, 30mpanl12 713 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
32313adant3 1073 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
33 simp3 1055 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
34 eqid 2606 . . . . . 6 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧)))
3522, 19, 34limcvallem 23355 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝑦 ∈ ℂ))
3635abssdv 3635 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ⊆ ℂ)
3729ssex 4722 . . . 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)
3836, 37syl 17 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ∈ V)
392, 28, 32, 33, 38ovmpt2d 6661 . 2 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐹 lim 𝐵) = {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)})
4039, 36eqsstrd 3598 . 2 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐹 lim 𝐵) ⊆ ℂ)
4139, 40jca 552 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 382  w3a 1030   = wceq 1474  wcel 1976  {cab 2592  Vcvv 3169  [wsbc 3398  cun 3534  wss 3536  ifcif 4032  {csn 4121  cmpt 4634  dom cdm 5025  wf 5783  cfv 5787  (class class class)co 6524  cmpt2 6526  pm cpm 7719  cc 9787  t crest 15847  TopOpenctopn 15848  fldccnfld 19510   CnP ccnp 20778   lim climc 23346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-map 7720  df-pm 7721  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-fi 8174  df-sup 8205  df-inf 8206  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-3 10924  df-4 10925  df-5 10926  df-6 10927  df-7 10928  df-8 10929  df-9 10930  df-n0 11137  df-z 11208  df-dec 11323  df-uz 11517  df-q 11618  df-rp 11662  df-xneg 11775  df-xadd 11776  df-xmul 11777  df-fz 12150  df-seq 12616  df-exp 12675  df-cj 13630  df-re 13631  df-im 13632  df-sqrt 13766  df-abs 13767  df-struct 15640  df-ndx 15641  df-slot 15642  df-base 15643  df-plusg 15724  df-mulr 15725  df-starv 15726  df-tset 15730  df-ple 15731  df-ds 15734  df-unif 15735  df-rest 15849  df-topn 15850  df-topgen 15870  df-psmet 19502  df-xmet 19503  df-met 19504  df-bl 19505  df-mopn 19506  df-cnfld 19511  df-top 20460  df-bases 20461  df-topon 20462  df-topsp 20463  df-cnp 20781  df-xms 21873  df-ms 21874  df-limc 23350
This theorem is referenced by:  ellimc  23357  limccl  23359
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