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Theorem ruclem13 16192
Description: Lemma for ruc 16193. There is no function that maps β„• onto ℝ. (Use nex 1794 if you want this in the form Β¬ βˆƒπ‘“π‘“:ℕ–onto→ℝ.) (Contributed by NM, 14-Oct-2004.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
Assertion
Ref Expression
ruclem13 Β¬ 𝐹:ℕ–onto→ℝ

Proof of Theorem ruclem13
Dummy variables π‘š 𝑑 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 forn 6802 . . . 4 (𝐹:ℕ–onto→ℝ β†’ ran 𝐹 = ℝ)
21difeq2d 4117 . . 3 (𝐹:ℕ–onto→ℝ β†’ (ℝ βˆ– ran 𝐹) = (ℝ βˆ– ℝ))
3 difid 4365 . . 3 (ℝ βˆ– ℝ) = βˆ…
42, 3eqtrdi 2782 . 2 (𝐹:ℕ–onto→ℝ β†’ (ℝ βˆ– ran 𝐹) = βˆ…)
5 reex 11203 . . . . . 6 ℝ ∈ V
65, 5xpex 7737 . . . . 5 (ℝ Γ— ℝ) ∈ V
76, 5mpoex 8065 . . . 4 (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)) ∈ V
87isseti 3484 . . 3 βˆƒπ‘‘ 𝑑 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩))
9 fof 6799 . . . . . . . 8 (𝐹:ℕ–onto→ℝ β†’ 𝐹:β„•βŸΆβ„)
109adantr 480 . . . . . . 7 ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩))) β†’ 𝐹:β„•βŸΆβ„)
11 simpr 484 . . . . . . 7 ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩))) β†’ 𝑑 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)))
12 eqid 2726 . . . . . . 7 ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹) = ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)
13 eqid 2726 . . . . . . 7 seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)) = seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹))
14 eqid 2726 . . . . . . 7 sup(ran (1st ∘ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹))), ℝ, < ) = sup(ran (1st ∘ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹))), ℝ, < )
1510, 11, 12, 13, 14ruclem12 16191 . . . . . 6 ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩))) β†’ sup(ran (1st ∘ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹))), ℝ, < ) ∈ (ℝ βˆ– ran 𝐹))
16 n0i 4328 . . . . . 6 (sup(ran (1st ∘ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹))), ℝ, < ) ∈ (ℝ βˆ– ran 𝐹) β†’ Β¬ (ℝ βˆ– ran 𝐹) = βˆ…)
1715, 16syl 17 . . . . 5 ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩))) β†’ Β¬ (ℝ βˆ– ran 𝐹) = βˆ…)
1817ex 412 . . . 4 (𝐹:ℕ–onto→ℝ β†’ (𝑑 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)) β†’ Β¬ (ℝ βˆ– ran 𝐹) = βˆ…))
1918exlimdv 1928 . . 3 (𝐹:ℕ–onto→ℝ β†’ (βˆƒπ‘‘ 𝑑 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)) β†’ Β¬ (ℝ βˆ– ran 𝐹) = βˆ…))
208, 19mpi 20 . 2 (𝐹:ℕ–onto→ℝ β†’ Β¬ (ℝ βˆ– ran 𝐹) = βˆ…)
214, 20pm2.65i 193 1 Β¬ 𝐹:ℕ–onto→ℝ
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 395   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  β¦‹csb 3888   βˆ– cdif 3940   βˆͺ cun 3941  βˆ…c0 4317  ifcif 4523  {csn 4623  βŸ¨cop 4629   class class class wbr 5141   Γ— cxp 5667  ran crn 5670   ∘ ccom 5673  βŸΆwf 6533  β€“ontoβ†’wfo 6535  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  1st c1st 7972  2nd c2nd 7973  supcsup 9437  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252   / cdiv 11875  β„•cn 12216  2c2 12271  seqcseq 13972
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-op 4630  df-uni 4903  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-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  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-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-seq 13973
This theorem is referenced by:  ruc  16193
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