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Theorem limcfval 25380
Description: Value and set bounds on the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
limcval.j 𝐽 = (𝐾 β†Ύt (𝐴 βˆͺ {𝐡}))
limcval.k 𝐾 = (TopOpenβ€˜β„‚fld)
Assertion
Ref Expression
limcfval ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐹 limβ„‚ 𝐡) = {𝑦 ∣ (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))) ∈ ((𝐽 CnP 𝐾)β€˜π΅)} ∧ (𝐹 limβ„‚ 𝐡) βŠ† β„‚))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐡,𝑧   𝑦,𝐹,𝑧   𝑦,𝐾,𝑧   𝑦,𝐽
Allowed substitution hint:   𝐽(𝑧)
Proof of Theorem limcfval
Dummy variables 𝑓 𝑗 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-limc 25374 . . . 4 limβ„‚ = (𝑓 ∈ (β„‚ ↑pm β„‚), π‘₯ ∈ β„‚ ↦ {𝑦 ∣ [(TopOpenβ€˜β„‚fld) / 𝑗](𝑧 ∈ (dom 𝑓 βˆͺ {π‘₯}) ↦ if(𝑧 = π‘₯, 𝑦, (π‘“β€˜π‘§))) ∈ (((𝑗 β†Ύt (dom 𝑓 βˆͺ {π‘₯})) CnP 𝑗)β€˜π‘₯)})
21a1i 11 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ limβ„‚ = (𝑓 ∈ (β„‚ ↑pm β„‚), π‘₯ ∈ β„‚ ↦ {𝑦 ∣ [(TopOpenβ€˜β„‚fld) / 𝑗](𝑧 ∈ (dom 𝑓 βˆͺ {π‘₯}) ↦ if(𝑧 = π‘₯, 𝑦, (π‘“β€˜π‘§))) ∈ (((𝑗 β†Ύt (dom 𝑓 βˆͺ {π‘₯})) CnP 𝑗)β€˜π‘₯)}))
3 fvexd 6903 . . . . 5 (((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) β†’ (TopOpenβ€˜β„‚fld) ∈ V)
4 simplrl 775 . . . . . . . . . 10 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ 𝑓 = 𝐹)
54dmeqd 5903 . . . . . . . . 9 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ dom 𝑓 = dom 𝐹)
6 simpll1 1212 . . . . . . . . . 10 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ 𝐹:π΄βŸΆβ„‚)
76fdmd 6725 . . . . . . . . 9 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ dom 𝐹 = 𝐴)
85, 7eqtrd 2772 . . . . . . . 8 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ dom 𝑓 = 𝐴)
9 simplrr 776 . . . . . . . . 9 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ π‘₯ = 𝐡)
109sneqd 4639 . . . . . . . 8 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ {π‘₯} = {𝐡})
118, 10uneq12d 4163 . . . . . . 7 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ (dom 𝑓 βˆͺ {π‘₯}) = (𝐴 βˆͺ {𝐡}))
129eqeq2d 2743 . . . . . . . 8 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ (𝑧 = π‘₯ ↔ 𝑧 = 𝐡))
134fveq1d 6890 . . . . . . . 8 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ (π‘“β€˜π‘§) = (πΉβ€˜π‘§))
1412, 13ifbieq2d 4553 . . . . . . 7 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ if(𝑧 = π‘₯, 𝑦, (π‘“β€˜π‘§)) = if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§)))
1511, 14mpteq12dv 5238 . . . . . 6 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ (𝑧 ∈ (dom 𝑓 βˆͺ {π‘₯}) ↦ if(𝑧 = π‘₯, 𝑦, (π‘“β€˜π‘§))) = (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))))
16 simpr 485 . . . . . . . . . . 11 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ 𝑗 = (TopOpenβ€˜β„‚fld))
17 limcval.k . . . . . . . . . . 11 𝐾 = (TopOpenβ€˜β„‚fld)
1816, 17eqtr4di 2790 . . . . . . . . . 10 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ 𝑗 = 𝐾)
1918, 11oveq12d 7423 . . . . . . . . 9 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ (𝑗 β†Ύt (dom 𝑓 βˆͺ {π‘₯})) = (𝐾 β†Ύt (𝐴 βˆͺ {𝐡})))
20 limcval.j . . . . . . . . 9 𝐽 = (𝐾 β†Ύt (𝐴 βˆͺ {𝐡}))
2119, 20eqtr4di 2790 . . . . . . . 8 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ (𝑗 β†Ύt (dom 𝑓 βˆͺ {π‘₯})) = 𝐽)
2221, 18oveq12d 7423 . . . . . . 7 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ ((𝑗 β†Ύt (dom 𝑓 βˆͺ {π‘₯})) CnP 𝑗) = (𝐽 CnP 𝐾))
2322, 9fveq12d 6895 . . . . . 6 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ (((𝑗 β†Ύt (dom 𝑓 βˆͺ {π‘₯})) CnP 𝑗)β€˜π‘₯) = ((𝐽 CnP 𝐾)β€˜π΅))
2415, 23eleq12d 2827 . . . . 5 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ ((𝑧 ∈ (dom 𝑓 βˆͺ {π‘₯}) ↦ if(𝑧 = π‘₯, 𝑦, (π‘“β€˜π‘§))) ∈ (((𝑗 β†Ύt (dom 𝑓 βˆͺ {π‘₯})) CnP 𝑗)β€˜π‘₯) ↔ (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))) ∈ ((𝐽 CnP 𝐾)β€˜π΅)))
253, 24sbcied 3821 . . . 4 (((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) β†’ ([(TopOpenβ€˜β„‚fld) / 𝑗](𝑧 ∈ (dom 𝑓 βˆͺ {π‘₯}) ↦ if(𝑧 = π‘₯, 𝑦, (π‘“β€˜π‘§))) ∈ (((𝑗 β†Ύt (dom 𝑓 βˆͺ {π‘₯})) CnP 𝑗)β€˜π‘₯) ↔ (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))) ∈ ((𝐽 CnP 𝐾)β€˜π΅)))
2625abbidv 2801 . . 3 (((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) β†’ {𝑦 ∣ [(TopOpenβ€˜β„‚fld) / 𝑗](𝑧 ∈ (dom 𝑓 βˆͺ {π‘₯}) ↦ if(𝑧 = π‘₯, 𝑦, (π‘“β€˜π‘§))) ∈ (((𝑗 β†Ύt (dom 𝑓 βˆͺ {π‘₯})) CnP 𝑗)β€˜π‘₯)} = {𝑦 ∣ (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))) ∈ ((𝐽 CnP 𝐾)β€˜π΅)})
27 cnex 11187 . . . . 5 β„‚ ∈ V
28 elpm2r 8835 . . . . 5 (((β„‚ ∈ V ∧ β„‚ ∈ V) ∧ (𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚)) β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
2927, 27, 28mpanl12 700 . . . 4 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚) β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
30293adant3 1132 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
31 simp3 1138 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ 𝐡 ∈ β„‚)
32 eqid 2732 . . . . . 6 (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))) = (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§)))
3320, 17, 32limcvallem 25379 . . . . 5 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))) ∈ ((𝐽 CnP 𝐾)β€˜π΅) β†’ 𝑦 ∈ β„‚))
3433abssdv 4064 . . . 4 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ {𝑦 ∣ (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))) ∈ ((𝐽 CnP 𝐾)β€˜π΅)} βŠ† β„‚)
3527ssex 5320 . . . 4 ({𝑦 ∣ (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))) ∈ ((𝐽 CnP 𝐾)β€˜π΅)} βŠ† β„‚ β†’ {𝑦 ∣ (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))) ∈ ((𝐽 CnP 𝐾)β€˜π΅)} ∈ V)
3634, 35syl 17 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ {𝑦 ∣ (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))) ∈ ((𝐽 CnP 𝐾)β€˜π΅)} ∈ V)
372, 26, 30, 31, 36ovmpod 7556 . 2 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐹 limβ„‚ 𝐡) = {𝑦 ∣ (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))) ∈ ((𝐽 CnP 𝐾)β€˜π΅)})
3837, 34eqsstrd 4019 . 2 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐹 limβ„‚ 𝐡) βŠ† β„‚)
3937, 38jca 512 1 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐹 limβ„‚ 𝐡) = {𝑦 ∣ (𝑧 ∈ (𝐴 βˆͺ {𝐡}) ↦ if(𝑧 = 𝐡, 𝑦, (πΉβ€˜π‘§))) ∈ ((𝐽 CnP 𝐾)β€˜π΅)} ∧ (𝐹 limβ„‚ 𝐡) βŠ† β„‚))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  Vcvv 3474  [wsbc 3776   βˆͺ cun 3945   βŠ† wss 3947  ifcif 4527  {csn 4627   ↦ cmpt 5230  dom cdm 5675  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407   ↑pm cpm 8817  β„‚cc 11104   β†Ύt crest 17362  TopOpenctopn 17363  β„‚fldccnfld 20936   CnP ccnp 22720   limβ„‚ climc 25370
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fi 9402  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-fz 13481  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-struct 17076  df-slot 17111  df-ndx 17123  df-base 17141  df-plusg 17206  df-mulr 17207  df-starv 17208  df-tset 17212  df-ple 17213  df-ds 17215  df-unif 17216  df-rest 17364  df-topn 17365  df-topgen 17385  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-cnfld 20937  df-top 22387  df-topon 22404  df-topsp 22426  df-bases 22440  df-cnp 22723  df-xms 23817  df-ms 23818  df-limc 25374
This theorem is referenced by:  ellimc  25381  limccl  25383
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