Theorem ruclem13 16218

Description: Lemma for ruc 16219. There is no function that maps ℕ onto ℝ. (Use nex 1794 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.

Step	Hyp	Ref	Expression
1		forn 6809	. . . 4 ⊢ (𝐹:ℕ–onto→ℝ → ran 𝐹 = ℝ)
2	1	difeq2d 4114	. . 3 ⊢ (𝐹:ℕ–onto→ℝ → (ℝ ∖ ran 𝐹) = (ℝ ∖ ℝ))
3		difid 4366	. . 3 ⊢ (ℝ ∖ ℝ) = ∅
4	2, 3	eqtrdi 2781	. 2 ⊢ (𝐹:ℕ–onto→ℝ → (ℝ ∖ ran 𝐹) = ∅)
5		reex 11229	. . . . . 6 ⊢ ℝ ∈ V
6	5, 5	xpex 7753	. . . . 5 ⊢ (ℝ × ℝ) ∈ V
7	6, 5	mpoex 8082	. . . 4 ⊢ (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩)) ∈ V
8	7	isseti 3479	. . 3 ⊢ ∃𝑑 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩))
9		fof 6806	. . . . . . . 8 ⊢ (𝐹:ℕ–onto→ℝ → 𝐹:ℕ⟶ℝ)
10	9	adantr 479	. . . . . . 7 ⊢ ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩))) → 𝐹:ℕ⟶ℝ)
11		simpr 483	. . . . . . 7 ⊢ ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩))) → 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩)))
12		eqid 2725	. . . . . . 7 ⊢ ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹) = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
13		eqid 2725	. . . . . . 7 ⊢ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)) = seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹))
14		eqid 2725	. . . . . . 7 ⊢ sup(ran (1st ∘ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹))), ℝ, < ) = sup(ran (1st ∘ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹))), ℝ, < )
15	10, 11, 12, 13, 14	ruclem12 16217	. . . . . 6 ⊢ ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩))) → sup(ran (1st ∘ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹))), ℝ, < ) ∈ (ℝ ∖ ran 𝐹))
16		n0i 4329	. . . . . 6 ⊢ (sup(ran (1st ∘ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹))), ℝ, < ) ∈ (ℝ ∖ ran 𝐹) → ¬ (ℝ ∖ ran 𝐹) = ∅)
17	15, 16	syl 17	. . . . 5 ⊢ ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩))) → ¬ (ℝ ∖ ran 𝐹) = ∅)
18	17	ex 411	. . . 4 ⊢ (𝐹:ℕ–onto→ℝ → (𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩)) → ¬ (ℝ ∖ ran 𝐹) = ∅))
19	18	exlimdv 1928	. . 3 ⊢ (𝐹:ℕ–onto→ℝ → (∃𝑑 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩)) → ¬ (ℝ ∖ ran 𝐹) = ∅))
20	8, 19	mpi 20	. 2 ⊢ (𝐹:ℕ–onto→ℝ → ¬ (ℝ ∖ ran 𝐹) = ∅)
21	4, 20	pm2.65i 193	1 ⊢ ¬ 𝐹:ℕ–onto→ℝ

Syntax hints:  ¬ wn 3   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ⦋csb 3884   ∖ cdif 3936   ∪ cun 3937  ∅c0 4318  ifcif 4524  {csn 4624  ⟨cop 4630   class class class wbr 5143   × cxp 5670  ran crn 5673   ∘ ccom 5676  ⟶wf 6539  –onto→wfo 6541  ‘cfv 6543  (class class class)co 7416   ∈ cmpo 7418  1^st c1st 7989  2^nd c2nd 7990  supcsup 9463  ℝcr 11137  0cc0 11138  1c1 11139   + caddc 11141   < clt 11278   / cdiv 11901  ℕcn 12242  2c2 12297  seqcseq 13998

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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-sup 9465  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-seq 13999

This theorem is referenced by:  ruc  16219