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Theorem limcfval 25756
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 25750 . . . 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 6900 . . . . 5 (((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) β†’ (TopOpenβ€˜β„‚fld) ∈ V)
4 simplrl 774 . . . . . . . . . 10 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ 𝑓 = 𝐹)
54dmeqd 5899 . . . . . . . . 9 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ dom 𝑓 = dom 𝐹)
6 simpll1 1209 . . . . . . . . . 10 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ 𝐹:π΄βŸΆβ„‚)
76fdmd 6722 . . . . . . . . 9 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ dom 𝐹 = 𝐴)
85, 7eqtrd 2766 . . . . . . . 8 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ dom 𝑓 = 𝐴)
9 simplrr 775 . . . . . . . . 9 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ π‘₯ = 𝐡)
109sneqd 4635 . . . . . . . 8 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ {π‘₯} = {𝐡})
118, 10uneq12d 4159 . . . . . . 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 2737 . . . . . . . 8 ((((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝑓 = 𝐹 ∧ π‘₯ = 𝐡)) ∧ 𝑗 = (TopOpenβ€˜β„‚fld)) β†’ (𝑧 = π‘₯ ↔ 𝑧 = 𝐡))
134fveq1d 6887 . . . . . . . 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 4549 . . . . . . 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 5232 . . . . . 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 484 . . . . . . . . . . 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 2784 . . . . . . . . . 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 2784 . . . . . . . 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 6892 . . . . . 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 2821 . . . . 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 3817 . . . 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 2795 . . 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 11193 . . . . 5 β„‚ ∈ V
28 elpm2r 8841 . . . . 5 (((β„‚ ∈ V ∧ β„‚ ∈ V) ∧ (𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚)) β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
2927, 27, 28mpanl12 699 . . . 4 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚) β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
30293adant3 1129 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
31 simp3 1135 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ 𝐡 ∈ β„‚)
32 eqid 2726 . . . . . 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 25755 . . . . 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 4060 . . . 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 5314 . . . 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 4015 . 2 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐹 limβ„‚ 𝐡) βŠ† β„‚)
3937, 38jca 511 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703  Vcvv 3468  [wsbc 3772   βˆͺ cun 3941   βŠ† wss 3943  ifcif 4523  {csn 4623   ↦ cmpt 5224  dom cdm 5669  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407   ↑pm cpm 8823  β„‚cc 11110   β†Ύt crest 17375  TopOpenctopn 17376  β„‚fldccnfld 21240   CnP ccnp 23084   limβ„‚ climc 25746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fi 9408  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-fz 13491  df-seq 13973  df-exp 14033  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-struct 17089  df-slot 17124  df-ndx 17136  df-base 17154  df-plusg 17219  df-mulr 17220  df-starv 17221  df-tset 17225  df-ple 17226  df-ds 17228  df-unif 17229  df-rest 17377  df-topn 17378  df-topgen 17398  df-psmet 21232  df-xmet 21233  df-met 21234  df-bl 21235  df-mopn 21236  df-cnfld 21241  df-top 22751  df-topon 22768  df-topsp 22790  df-bases 22804  df-cnp 23087  df-xms 24181  df-ms 24182  df-limc 25750
This theorem is referenced by:  ellimc  25757  limccl  25759
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