Theorem ruclem13 16278

Description: Lemma for ruc 16279. There is no function that maps ℕ onto ℝ. (Use nex 1800 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 6823	. . . 4 ⊢ (𝐹:ℕ–onto→ℝ → ran 𝐹 = ℝ)
2	1	difeq2d 4126	. . 3 ⊢ (𝐹:ℕ–onto→ℝ → (ℝ ∖ ran 𝐹) = (ℝ ∖ ℝ))
3		difid 4376	. . 3 ⊢ (ℝ ∖ ℝ) = ∅
4	2, 3	eqtrdi 2793	. 2 ⊢ (𝐹:ℕ–onto→ℝ → (ℝ ∖ ran 𝐹) = ∅)
5		reex 11246	. . . . . 6 ⊢ ℝ ∈ V
6	5, 5	xpex 7773	. . . . 5 ⊢ (ℝ × ℝ) ∈ V
7	6, 5	mpoex 8104	. . . 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 3498	. . 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 6820	. . . . . . . 8 ⊢ (𝐹:ℕ–onto→ℝ → 𝐹:ℕ⟶ℝ)
10	9	adantr 480	. . . . . . 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 484	. . . . . . 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 2737	. . . . . . 7 ⊢ ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹) = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
13		eqid 2737	. . . . . . 7 ⊢ seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)) = seq0(𝑑, ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹))
14		eqid 2737	. . . . . . 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 16277	. . . . . 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 4340	. . . . . 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 412	. . . 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 1933	. . 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 194	1 ⊢ ¬ 𝐹:ℕ–onto→ℝ

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ⦋csb 3899   ∖ cdif 3948   ∪ cun 3949  ∅c0 4333  ifcif 4525  {csn 4626  ⟨cop 4632   class class class wbr 5143   × cxp 5683  ran crn 5686   ∘ ccom 5689  ⟶wf 6557  –onto→wfo 6559  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  1^st c1st 8012  2^nd c2nd 8013  supcsup 9480  ℝcr 11154  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295   / cdiv 11920  ℕcn 12266  2c2 12321  seqcseq 14042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-seq 14043

This theorem is referenced by:  ruc  16279