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Mirrors > Home > MPE Home > Th. List > om2uzf1oi | Structured version Visualization version GIF version |
Description: 𝐺 (see om2uz0i 13967) 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 8465 | . . . . 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 6657 | . . . . 5 ⊢ (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) Fn ω) |
4 | 1, 3 | mpbir 230 | . . . 4 ⊢ 𝐺 Fn ω |
5 | om2uz.1 | . . . . . 6 ⊢ 𝐶 ∈ ℤ | |
6 | 5, 2 | om2uzrani 13972 | . . . . 5 ⊢ ran 𝐺 = (ℤ≥‘𝐶) |
7 | 6 | eqimssi 4040 | . . . 4 ⊢ ran 𝐺 ⊆ (ℤ≥‘𝐶) |
8 | df-f 6558 | . . . 4 ⊢ (𝐺:ω⟶(ℤ≥‘𝐶) ↔ (𝐺 Fn ω ∧ ran 𝐺 ⊆ (ℤ≥‘𝐶))) | |
9 | 4, 7, 8 | mpbir2an 709 | . . 3 ⊢ 𝐺:ω⟶(ℤ≥‘𝐶) |
10 | 5, 2 | om2uzuzi 13969 | . . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ (ℤ≥‘𝐶)) |
11 | eluzelz 12884 | . . . . . . . . 9 ⊢ ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) → (𝐺‘𝑦) ∈ ℤ) | |
12 | 10, 11 | syl 17 | . . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℤ) |
13 | 12 | zred 12718 | . . . . . . 7 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℝ) |
14 | 5, 2 | om2uzuzi 13969 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ (ℤ≥‘𝐶)) |
15 | eluzelz 12884 | . . . . . . . . 9 ⊢ ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) → (𝐺‘𝑧) ∈ ℤ) | |
16 | 14, 15 | syl 17 | . . . . . . . 8 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℤ) |
17 | 16 | zred 12718 | . . . . . . 7 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℝ) |
18 | lttri3 11347 | . . . . . . 7 ⊢ (((𝐺‘𝑦) ∈ ℝ ∧ (𝐺‘𝑧) ∈ ℝ) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ (¬ (𝐺‘𝑦) < (𝐺‘𝑧) ∧ ¬ (𝐺‘𝑧) < (𝐺‘𝑦)))) | |
19 | 13, 17, 18 | syl2an 594 | . . . . . 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 | bitr4di 288 | . . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ ¬ ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)))) |
22 | nnord 7884 | . . . . . . . . 9 ⊢ (𝑦 ∈ ω → Ord 𝑦) | |
23 | nnord 7884 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → Ord 𝑧) | |
24 | ordtri3 6412 | . . . . . . . . 9 ⊢ ((Ord 𝑦 ∧ Ord 𝑧) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))) | |
25 | 22, 23, 24 | syl2an 594 | . . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))) |
26 | 25 | con2bid 353 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦) ↔ ¬ 𝑦 = 𝑧)) |
27 | 5, 2 | om2uzlti 13970 | . . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 ∈ 𝑧 → (𝐺‘𝑦) < (𝐺‘𝑧))) |
28 | 5, 2 | om2uzlti 13970 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ω ∧ 𝑦 ∈ ω) → (𝑧 ∈ 𝑦 → (𝐺‘𝑧) < (𝐺‘𝑦))) |
29 | 28 | ancoms 457 | . . . . . . . 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 259 | . . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (¬ 𝑦 = 𝑧 → ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)))) |
32 | 31 | con1d 145 | . . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (¬ ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)) → 𝑦 = 𝑧)) |
33 | 21, 32 | sylbid 239 | . . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)) |
34 | 33 | rgen2 3188 | . . 3 ⊢ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧) |
35 | dff13 7270 | . . 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 709 | . 2 ⊢ 𝐺:ω–1-1→(ℤ≥‘𝐶) |
37 | dff1o5 6852 | . 2 ⊢ (𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ↔ (𝐺:ω–1-1→(ℤ≥‘𝐶) ∧ ran 𝐺 = (ℤ≥‘𝐶))) | |
38 | 36, 6, 37 | mpbir2an 709 | 1 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1534 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ⊆ wss 3947 class class class wbr 5153 ↦ cmpt 5236 ran crn 5683 ↾ cres 5684 Ord word 6375 Fn wfn 6549 ⟶wf 6550 –1-1→wf1 6551 –1-1-onto→wf1o 6553 ‘cfv 6554 (class class class)co 7424 ωcom 7876 reccrdg 8439 ℝcr 11157 1c1 11159 + caddc 11161 < clt 11298 ℤcz 12610 ℤ≥cuz 12874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 |
This theorem is referenced by: om2uzisoi 13974 uzrdglem 13977 uzrdgfni 13978 uzrdgsuci 13980 uzenom 13984 fzennn 13988 cardfz 13990 hashgf1o 13991 axdc4uzlem 14003 unbenlem 16910 |
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