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Mirrors > Home > MPE Home > Th. List > om2uzf1oi | Structured version Visualization version GIF version |
Description: 𝐺 (see om2uz0i 13310) 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 8053 | . . . . 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 6420 | . . . . 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 13315 | . . . . 5 ⊢ ran 𝐺 = (ℤ≥‘𝐶) |
7 | 6 | eqimssi 3973 | . . . 4 ⊢ ran 𝐺 ⊆ (ℤ≥‘𝐶) |
8 | df-f 6328 | . . . 4 ⊢ (𝐺:ω⟶(ℤ≥‘𝐶) ↔ (𝐺 Fn ω ∧ ran 𝐺 ⊆ (ℤ≥‘𝐶))) | |
9 | 4, 7, 8 | mpbir2an 710 | . . 3 ⊢ 𝐺:ω⟶(ℤ≥‘𝐶) |
10 | 5, 2 | om2uzuzi 13312 | . . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ (ℤ≥‘𝐶)) |
11 | eluzelz 12241 | . . . . . . . . 9 ⊢ ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) → (𝐺‘𝑦) ∈ ℤ) | |
12 | 10, 11 | syl 17 | . . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℤ) |
13 | 12 | zred 12075 | . . . . . . 7 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℝ) |
14 | 5, 2 | om2uzuzi 13312 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ (ℤ≥‘𝐶)) |
15 | eluzelz 12241 | . . . . . . . . 9 ⊢ ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) → (𝐺‘𝑧) ∈ ℤ) | |
16 | 14, 15 | syl 17 | . . . . . . . 8 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℤ) |
17 | 16 | zred 12075 | . . . . . . 7 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℝ) |
18 | lttri3 10713 | . . . . . . 7 ⊢ (((𝐺‘𝑦) ∈ ℝ ∧ (𝐺‘𝑧) ∈ ℝ) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ (¬ (𝐺‘𝑦) < (𝐺‘𝑧) ∧ ¬ (𝐺‘𝑧) < (𝐺‘𝑦)))) | |
19 | 13, 17, 18 | syl2an 598 | . . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ (¬ (𝐺‘𝑦) < (𝐺‘𝑧) ∧ ¬ (𝐺‘𝑧) < (𝐺‘𝑦)))) |
20 | ioran 981 | . . . . . 6 ⊢ (¬ ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)) ↔ (¬ (𝐺‘𝑦) < (𝐺‘𝑧) ∧ ¬ (𝐺‘𝑧) < (𝐺‘𝑦))) | |
21 | 19, 20 | syl6bbr 292 | . . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ ¬ ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)))) |
22 | nnord 7568 | . . . . . . . . 9 ⊢ (𝑦 ∈ ω → Ord 𝑦) | |
23 | nnord 7568 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → Ord 𝑧) | |
24 | ordtri3 6195 | . . . . . . . . 9 ⊢ ((Ord 𝑦 ∧ Ord 𝑧) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))) | |
25 | 22, 23, 24 | syl2an 598 | . . . . . . . 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 13313 | . . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 ∈ 𝑧 → (𝐺‘𝑦) < (𝐺‘𝑧))) |
28 | 5, 2 | om2uzlti 13313 | . . . . . . . . 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 962 | . . . . . . 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 3168 | . . 3 ⊢ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧) |
35 | dff13 6991 | . . 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 710 | . 2 ⊢ 𝐺:ω–1-1→(ℤ≥‘𝐶) |
37 | dff1o5 6599 | . 2 ⊢ (𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ↔ (𝐺:ω–1-1→(ℤ≥‘𝐶) ∧ ran 𝐺 = (ℤ≥‘𝐶))) | |
38 | 36, 6, 37 | mpbir2an 710 | 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 844 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 ↾ cres 5521 Ord word 6158 Fn wfn 6319 ⟶wf 6320 –1-1→wf1 6321 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ωcom 7560 reccrdg 8028 ℝcr 10525 1c1 10527 + caddc 10529 < clt 10664 ℤcz 11969 ℤ≥cuz 12231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 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 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 |
This theorem is referenced by: om2uzisoi 13317 uzrdglem 13320 uzrdgfni 13321 uzrdgsuci 13323 uzenom 13327 fzennn 13331 cardfz 13333 hashgf1o 13334 axdc4uzlem 13346 unbenlem 16234 |
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