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Mirrors > Home > MPE Home > Th. List > om2uzisoi | Structured version Visualization version GIF version |
Description: 𝐺 (see om2uz0i 12746) is an isomorphism from natural ordinals to upper integers. (Contributed by NM, 9-Oct-2008.) (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 |
---|---|
om2uzisoi | ⊢ 𝐺 Isom E , < (ω, (ℤ≥‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | om2uz.1 | . . 3 ⊢ 𝐶 ∈ ℤ | |
2 | om2uz.2 | . . 3 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) | |
3 | 1, 2 | om2uzf1oi 12752 | . 2 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶) |
4 | epel 5032 | . . . 4 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
5 | 1, 2 | om2uzlt2i 12750 | . . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 ∈ 𝑧 ↔ (𝐺‘𝑦) < (𝐺‘𝑧))) |
6 | 4, 5 | syl5bb 272 | . . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 E 𝑧 ↔ (𝐺‘𝑦) < (𝐺‘𝑧))) |
7 | 6 | rgen2a 2977 | . 2 ⊢ ∀𝑦 ∈ ω ∀𝑧 ∈ ω (𝑦 E 𝑧 ↔ (𝐺‘𝑦) < (𝐺‘𝑧)) |
8 | df-isom 5897 | . 2 ⊢ (𝐺 Isom E , < (ω, (ℤ≥‘𝐶)) ↔ (𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ ∀𝑦 ∈ ω ∀𝑧 ∈ ω (𝑦 E 𝑧 ↔ (𝐺‘𝑦) < (𝐺‘𝑧)))) | |
9 | 3, 7, 8 | mpbir2an 955 | 1 ⊢ 𝐺 Isom E , < (ω, (ℤ≥‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 class class class wbr 4653 ↦ cmpt 4729 E cep 5028 ↾ cres 5116 –1-1-onto→wf1o 5887 ‘cfv 5888 Isom wiso 5889 (class class class)co 6650 ωcom 7065 reccrdg 7505 1c1 9937 + caddc 9939 < clt 10074 ℤcz 11377 ℤ≥cuz 11687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 |
This theorem is referenced by: om2uzoi 12754 ltweuz 12760 fz1isolem 13245 |
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