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Mirrors > Home > MPE Home > Th. List > om2uzf1oi | Structured version Visualization version GIF version |
Description: 𝐺 (see om2uz0i 13399) is a one-to-one onto mapping. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Ref | Expression |
---|---|
om2uz.1 | ⊢ 𝐶 ∈ ℤ |
om2uz.2 | ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) |
Ref | Expression |
---|---|
om2uzf1oi | ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frfnom 8092 | . . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) Fn ω | |
2 | om2uz.2 | . . . . . 6 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) | |
3 | 2 | fneq1i 6429 | . . . . 5 ⊢ (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) Fn ω) |
4 | 1, 3 | mpbir 234 | . . . 4 ⊢ 𝐺 Fn ω |
5 | om2uz.1 | . . . . . 6 ⊢ 𝐶 ∈ ℤ | |
6 | 5, 2 | om2uzrani 13404 | . . . . 5 ⊢ ran 𝐺 = (ℤ≥‘𝐶) |
7 | 6 | eqimssi 3933 | . . . 4 ⊢ ran 𝐺 ⊆ (ℤ≥‘𝐶) |
8 | df-f 6337 | . . . 4 ⊢ (𝐺:ω⟶(ℤ≥‘𝐶) ↔ (𝐺 Fn ω ∧ ran 𝐺 ⊆ (ℤ≥‘𝐶))) | |
9 | 4, 7, 8 | mpbir2an 711 | . . 3 ⊢ 𝐺:ω⟶(ℤ≥‘𝐶) |
10 | 5, 2 | om2uzuzi 13401 | . . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ (ℤ≥‘𝐶)) |
11 | eluzelz 12327 | . . . . . . . . 9 ⊢ ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) → (𝐺‘𝑦) ∈ ℤ) | |
12 | 10, 11 | syl 17 | . . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℤ) |
13 | 12 | zred 12161 | . . . . . . 7 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℝ) |
14 | 5, 2 | om2uzuzi 13401 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ (ℤ≥‘𝐶)) |
15 | eluzelz 12327 | . . . . . . . . 9 ⊢ ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) → (𝐺‘𝑧) ∈ ℤ) | |
16 | 14, 15 | syl 17 | . . . . . . . 8 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℤ) |
17 | 16 | zred 12161 | . . . . . . 7 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℝ) |
18 | lttri3 10795 | . . . . . . 7 ⊢ (((𝐺‘𝑦) ∈ ℝ ∧ (𝐺‘𝑧) ∈ ℝ) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ (¬ (𝐺‘𝑦) < (𝐺‘𝑧) ∧ ¬ (𝐺‘𝑧) < (𝐺‘𝑦)))) | |
19 | 13, 17, 18 | syl2an 599 | . . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ (¬ (𝐺‘𝑦) < (𝐺‘𝑧) ∧ ¬ (𝐺‘𝑧) < (𝐺‘𝑦)))) |
20 | ioran 983 | . . . . . 6 ⊢ (¬ ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)) ↔ (¬ (𝐺‘𝑦) < (𝐺‘𝑧) ∧ ¬ (𝐺‘𝑧) < (𝐺‘𝑦))) | |
21 | 19, 20 | bitr4di 292 | . . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ ¬ ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)))) |
22 | nnord 7601 | . . . . . . . . 9 ⊢ (𝑦 ∈ ω → Ord 𝑦) | |
23 | nnord 7601 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → Ord 𝑧) | |
24 | ordtri3 6202 | . . . . . . . . 9 ⊢ ((Ord 𝑦 ∧ Ord 𝑧) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))) | |
25 | 22, 23, 24 | syl2an 599 | . . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))) |
26 | 25 | con2bid 358 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦) ↔ ¬ 𝑦 = 𝑧)) |
27 | 5, 2 | om2uzlti 13402 | . . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 ∈ 𝑧 → (𝐺‘𝑦) < (𝐺‘𝑧))) |
28 | 5, 2 | om2uzlti 13402 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ω ∧ 𝑦 ∈ ω) → (𝑧 ∈ 𝑦 → (𝐺‘𝑧) < (𝐺‘𝑦))) |
29 | 28 | ancoms 462 | . . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑧 ∈ 𝑦 → (𝐺‘𝑧) < (𝐺‘𝑦))) |
30 | 27, 29 | orim12d 964 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦) → ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)))) |
31 | 26, 30 | sylbird 263 | . . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (¬ 𝑦 = 𝑧 → ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)))) |
32 | 31 | con1d 147 | . . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (¬ ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)) → 𝑦 = 𝑧)) |
33 | 21, 32 | sylbid 243 | . . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)) |
34 | 33 | rgen2 3115 | . . 3 ⊢ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧) |
35 | dff13 7018 | . . 3 ⊢ (𝐺:ω–1-1→(ℤ≥‘𝐶) ↔ (𝐺:ω⟶(ℤ≥‘𝐶) ∧ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧))) | |
36 | 9, 34, 35 | mpbir2an 711 | . 2 ⊢ 𝐺:ω–1-1→(ℤ≥‘𝐶) |
37 | dff1o5 6621 | . 2 ⊢ (𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ↔ (𝐺:ω–1-1→(ℤ≥‘𝐶) ∧ ran 𝐺 = (ℤ≥‘𝐶))) | |
38 | 36, 6, 37 | mpbir2an 711 | 1 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2113 ∀wral 3053 Vcvv 3397 ⊆ wss 3841 class class class wbr 5027 ↦ cmpt 5107 ran crn 5520 ↾ cres 5521 Ord word 6165 Fn wfn 6328 ⟶wf 6329 –1-1→wf1 6330 –1-1-onto→wf1o 6332 ‘cfv 6333 (class class class)co 7164 ωcom 7593 reccrdg 8067 ℝcr 10607 1c1 10609 + caddc 10611 < clt 10746 ℤcz 12055 ℤ≥cuz 12317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-n0 11970 df-z 12056 df-uz 12318 |
This theorem is referenced by: om2uzisoi 13406 uzrdglem 13409 uzrdgfni 13410 uzrdgsuci 13412 uzenom 13416 fzennn 13420 cardfz 13422 hashgf1o 13423 axdc4uzlem 13435 unbenlem 16337 |
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