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Theorem ruclem13 15191
Description: Lemma for ruc 15192. There is no function that maps onto . (Use nex 1880 if you want this in the form ¬ ∃𝑓𝑓:ℕ–onto→ℝ.) (Contributed by NM, 14-Oct-2004.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
Assertion
Ref Expression
ruclem13 ¬ 𝐹:ℕ–onto→ℝ

Proof of Theorem ruclem13
Dummy variables 𝑚 𝑑 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 forn 6281 . . . 4 (𝐹:ℕ–onto→ℝ → ran 𝐹 = ℝ)
21difeq2d 3872 . . 3 (𝐹:ℕ–onto→ℝ → (ℝ ∖ ran 𝐹) = (ℝ ∖ ℝ))
3 difid 4092 . . 3 (ℝ ∖ ℝ) = ∅
42, 3syl6eq 2811 . 2 (𝐹:ℕ–onto→ℝ → (ℝ ∖ ran 𝐹) = ∅)
5 reex 10240 . . . . . 6 ℝ ∈ V
65, 5xpex 7129 . . . . 5 (ℝ × ℝ) ∈ V
76, 5mpt2ex 7417 . . . 4 (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)) ∈ V
87isseti 3350 . . 3 𝑑 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩))
9 fof 6278 . . . . . . . 8 (𝐹:ℕ–onto→ℝ → 𝐹:ℕ⟶ℝ)
109adantr 472 . . . . . . 7 ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩))) → 𝐹:ℕ⟶ℝ)
11 simpr 479 . . . . . . 7 ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩))) → 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
12 eqid 2761 . . . . . . 7 ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹) = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
13 eqid 2761 . . . . . . 7 seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)) = seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹))
14 eqid 2761 . . . . . . 7 sup(ran (1st ∘ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹))), ℝ, < ) = sup(ran (1st ∘ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹))), ℝ, < )
1510, 11, 12, 13, 14ruclem12 15190 . . . . . 6 ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩))) → sup(ran (1st ∘ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹))), ℝ, < ) ∈ (ℝ ∖ ran 𝐹))
16 n0i 4064 . . . . . 6 (sup(ran (1st ∘ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹))), ℝ, < ) ∈ (ℝ ∖ ran 𝐹) → ¬ (ℝ ∖ ran 𝐹) = ∅)
1715, 16syl 17 . . . . 5 ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩))) → ¬ (ℝ ∖ ran 𝐹) = ∅)
1817ex 449 . . . 4 (𝐹:ℕ–onto→ℝ → (𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)) → ¬ (ℝ ∖ ran 𝐹) = ∅))
1918exlimdv 2011 . . 3 (𝐹:ℕ–onto→ℝ → (∃𝑑 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)) → ¬ (ℝ ∖ ran 𝐹) = ∅))
208, 19mpi 20 . 2 (𝐹:ℕ–onto→ℝ → ¬ (ℝ ∖ ran 𝐹) = ∅)
214, 20pm2.65i 185 1 ¬ 𝐹:ℕ–onto→ℝ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1632  wex 1853  wcel 2140  csb 3675  cdif 3713  cun 3714  c0 4059  ifcif 4231  {csn 4322  cop 4328   class class class wbr 4805   × cxp 5265  ran crn 5268  ccom 5271  wf 6046  ontowfo 6048  cfv 6050  (class class class)co 6815  cmpt2 6817  1st c1st 7333  2nd c2nd 7334  supcsup 8514  cr 10148  0cc0 10149  1c1 10150   + caddc 10152   < clt 10287   / cdiv 10897  cn 11233  2c2 11283  seqcseq 13016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226  ax-pre-sup 10227
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-er 7914  df-en 8125  df-dom 8126  df-sdom 8127  df-sup 8516  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-div 10898  df-nn 11234  df-2 11292  df-n0 11506  df-z 11591  df-uz 11901  df-fz 12541  df-seq 13017
This theorem is referenced by:  ruc  15192
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