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Theorem ruclem13 16131
Description: Lemma for ruc 16132. There is no function that maps β„• onto ℝ. (Use nex 1803 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 6764 . . . 4 (𝐹:ℕ–onto→ℝ β†’ ran 𝐹 = ℝ)
21difeq2d 4087 . . 3 (𝐹:ℕ–onto→ℝ β†’ (ℝ βˆ– ran 𝐹) = (ℝ βˆ– ℝ))
3 difid 4335 . . 3 (ℝ βˆ– ℝ) = βˆ…
42, 3eqtrdi 2793 . 2 (𝐹:ℕ–onto→ℝ β†’ (ℝ βˆ– ran 𝐹) = βˆ…)
5 reex 11149 . . . . . 6 ℝ ∈ V
65, 5xpex 7692 . . . . 5 (ℝ Γ— ℝ) ∈ V
76, 5mpoex 8017 . . . 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 3463 . . 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 6761 . . . . . . . 8 (𝐹:ℕ–onto→ℝ β†’ 𝐹:β„•βŸΆβ„)
109adantr 482 . . . . . . 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 486 . . . . . . 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 2737 . . . . . . 7 ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹) = ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)
13 eqid 2737 . . . . . . 7 seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)) = seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹))
14 eqid 2737 . . . . . . 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 16130 . . . . . 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 4298 . . . . . 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 414 . . . 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 1937 . . 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 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  β¦‹csb 3860   βˆ– cdif 3912   βˆͺ cun 3913  βˆ…c0 4287  ifcif 4491  {csn 4591  βŸ¨cop 4597   class class class wbr 5110   Γ— cxp 5636  ran crn 5639   ∘ ccom 5642  βŸΆwf 6497  β€“ontoβ†’wfo 6499  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364  1st c1st 7924  2nd c2nd 7925  supcsup 9383  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   < clt 11196   / cdiv 11819  β„•cn 12160  2c2 12215  seqcseq 13913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9385  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-seq 13914
This theorem is referenced by:  ruc  16132
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