Theorem om2uzf1o 4656

Description: G (see om2uz0 4651) is a one-to-one onto mapping.

Hypotheses

Ref	Expression
om2uz.1	⊢ C ∈ ℤ
om2uz.2	⊢ G = (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω)

Assertion

Ref	Expression
om2uzf1o	⊢ G:ω–1-1-onto→{z ∈ ℤ∣C ≤ z}

Distinct variable group(s):   x,y,z   z,G   x,C,y,z

Proof of Theorem om2uzf1o

Step	Hyp	Ref	Expression
1		frfnom 2989	. . . . . . . 8 ⊢ (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) Fn ω
2		om2uz.2	. . . . . . . . 9 ⊢ G = (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω)
3		fneq1 2718	. . . . . . . . 9 ⊢ (G = (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) → (G Fn ω ↔ (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) Fn ω))
4	2, 3	ax-mp 6	. . . . . . . 8 ⊢ (G Fn ω ↔ (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) Fn ω)
5	1, 4	mpbir 165	. . . . . . 7 ⊢ G Fn ω
6		om2uz.1	. . . . . . . . 9 ⊢ C ∈ ℤ
7	6, 2	om2uzran 4655	. . . . . . . 8 ⊢ ran G = {z ∈ ℤ∣C ≤ z}
8		eqimss 1548	. . . . . . . 8 ⊢ (ran G = {z ∈ ℤ∣C ≤ z} → ran G ⊆ {z ∈ ℤ∣C ≤ z})
9	7, 8	ax-mp 6	. . . . . . 7 ⊢ ran G ⊆ {z ∈ ℤ∣C ≤ z}
10	5, 9	pm3.2i 234	. . . . . 6 ⊢ (G Fn ω ∧ ran G ⊆ {z ∈ ℤ∣C ≤ z})
11		df-f 2434	. . . . . 6 ⊢ (G:ω–→{z ∈ ℤ∣C ≤ z} ↔ (G Fn ω ∧ ran G ⊆ {z ∈ ℤ∣C ≤ z}))
12	10, 11	mpbir 165	. . . . 5 ⊢ G:ω–→{z ∈ ℤ∣C ≤ z}
13		lttri3t 4281	. . . . . . . . 9 ⊢ (((G ‘w) ∈ ℝ ∧ (G ‘v) ∈ ℝ) → ((G ‘w) = (G ‘v) ↔ (¬ (G ‘w) < (G ‘v) ∧ ¬ (G ‘v) < (G ‘w))))
14	6, 2	om2uzuz 4653	. . . . . . . . . 10 ⊢ (w ∈ ω → (G ‘w) ∈ {z ∈ ℤ∣C ≤ z})
15		ssrab 1556	. . . . . . . . . . 11 ⊢ {z ∈ ℤ∣C ≤ z} ⊆ ℤ
16	15	sseli 1504	. . . . . . . . . 10 ⊢ ((G ‘w) ∈ {z ∈ ℤ∣C ≤ z} → (G ‘w) ∈ ℤ)
17		zret 4567	. . . . . . . . . 10 ⊢ ((G ‘w) ∈ ℤ → (G ‘w) ∈ ℝ)
18	14, 16, 17	3syl 21	. . . . . . . . 9 ⊢ (w ∈ ω → (G ‘w) ∈ ℝ)
19	6, 2	om2uzuz 4653	. . . . . . . . . 10 ⊢ (v ∈ ω → (G ‘v) ∈ {z ∈ ℤ∣C ≤ z})
20	15	sseli 1504	. . . . . . . . . 10 ⊢ ((G ‘v) ∈ {z ∈ ℤ∣C ≤ z} → (G ‘v) ∈ ℤ)
21		zret 4567	. . . . . . . . . 10 ⊢ ((G ‘v) ∈ ℤ → (G ‘v) ∈ ℝ)
22	19, 20, 21	3syl 21	. . . . . . . . 9 ⊢ (v ∈ ω → (G ‘v) ∈ ℝ)
23	13, 18, 22	syl2an 349	. . . . . . . 8 ⊢ ((w ∈ ω ∧ v ∈ ω) → ((G ‘w) = (G ‘v) ↔ (¬ (G ‘w) < (G ‘v) ∧ ¬ (G ‘v) < (G ‘w))))
24		ioran 254	. . . . . . . 8 ⊢ (¬ ((G ‘w) < (G ‘v) ∨ (G ‘v) < (G ‘w)) ↔ (¬ (G ‘w) < (G ‘v) ∧ ¬ (G ‘v) < (G ‘w)))
25	23, 24	syl6bbr 416	. . . . . . 7 ⊢ ((w ∈ ω ∧ v ∈ ω) → ((G ‘w) = (G ‘v) ↔ ¬ ((G ‘w) < (G ‘v) ∨ (G ‘v) < (G ‘w))))
26		ordtri3 2234	. . . . . . . . . . 11 ⊢ ((Ord w ∧ Ord v) → (w = v ↔ ¬ (w ∈ v ∨ v ∈ w)))
27		nnord 2381	. . . . . . . . . . 11 ⊢ (w ∈ ω → Ord w)
28		nnord 2381	. . . . . . . . . . 11 ⊢ (v ∈ ω → Ord v)
29	26, 27, 28	syl2an 349	. . . . . . . . . 10 ⊢ ((w ∈ ω ∧ v ∈ ω) → (w = v ↔ ¬ (w ∈ v ∨ v ∈ w)))
30	29	bicon2d 404	. . . . . . . . 9 ⊢ ((w ∈ ω ∧ v ∈ ω) → ((w ∈ v ∨ v ∈ w) ↔ ¬ w = v))
31	6, 2	om2uzlt 4654	. . . . . . . . . 10 ⊢ ((w ∈ ω ∧ v ∈ ω) → (w ∈ v → (G ‘w) < (G ‘v)))
32	6, 2	om2uzlt 4654	. . . . . . . . . . 11 ⊢ ((v ∈ ω ∧ w ∈ ω) → (v ∈ w → (G ‘v) < (G ‘w)))
33	32	ancoms 334	. . . . . . . . . 10 ⊢ ((w ∈ ω ∧ v ∈ ω) → (v ∈ w → (G ‘v) < (G ‘w)))
34	31, 33	orim12d 436	. . . . . . . . 9 ⊢ ((w ∈ ω ∧ v ∈ ω) → ((w ∈ v ∨ v ∈ w) → ((G ‘w) < (G ‘v) ∨ (G ‘v) < (G ‘w))))
35	30, 34	sylbird 180	. . . . . . . 8 ⊢ ((w ∈ ω ∧ v ∈ ω) → (¬ w = v → ((G ‘w) < (G ‘v) ∨ (G ‘v) < (G ‘w))))
36	35	con1d 85	. . . . . . 7 ⊢ ((w ∈ ω ∧ v ∈ ω) → (¬ ((G ‘w) < (G ‘v) ∨ (G ‘v) < (G ‘w)) → w = v))
37	25, 36	sylbid 178	. . . . . 6 ⊢ ((w ∈ ω ∧ v ∈ ω) → ((G ‘w) = (G ‘v) → w = v))
38	37	rgen2 1248	. . . . 5 ⊢ ∀w ∈ ω ∀v ∈ ω ((G ‘w) = (G ‘v) → w = v)
39	12, 38	pm3.2i 234	. . . 4 ⊢ (G:ω–→{z ∈ ℤ∣C ≤ z} ∧ ∀w ∈ ω ∀v ∈ ω ((G ‘w) = (G ‘v) → w = v))
40		f1fv 2916	. . . 4 ⊢ (G:ω–1-1→{z ∈ ℤ∣C ≤ z} ↔ (G:ω–→{z ∈ ℤ∣C ≤ z} ∧ ∀w ∈ ω ∀v ∈ ω ((G ‘w) = (G ‘v) → w = v)))
41	39, 40	mpbir 165	. . 3 ⊢ G:ω–1-1→{z ∈ ℤ∣C ≤ z}
42	41, 7	pm3.2i 234	. 2 ⊢ (G:ω–1-1→{z ∈ ℤ∣C ≤ z} ∧ ran G = {z ∈ ℤ∣C ≤ z})
43		f1o5 2808	. 2 ⊢ (G:ω–1-1-onto→{z ∈ ℤ∣C ≤ z} ↔ (G:ω–1-1→{z ∈ ℤ∣C ≤ z} ∧ ran G = {z ∈ ℤ∣C ≤ z}))
44	42, 43	mpbir 165	1 ⊢ G:ω–1-1-onto→{z ∈ ℤ∣C ≤ z}

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = weq 797   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204   ⊆ wss 1487   class class class wbr 2054  {copab 2055  Ord word 2198  ωcom 2372  ran crn 2411   ↾ cres 2412   Fn wfn 2417  –→wf 2418  –1-1→wf1 2419  –1-1-onto→wf1o 2421   ‘cfv 2422  reccrdg 2969  (class class class)co 3001  ℝcr 4027  1c1 4029   + caddc 4031   < clt 4033   ≤ cle 4092  ℤcz 4095

This theorem is referenced by:  uzrdgval 4657  uzrdgini 4658  uzrdgsuc 4659  nnenom 4926

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-le 4277  df-n 4423  df-n0 4535  df-z 4564

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