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Mirrors > Home > MPE Home > Th. List > om2uzisoi | Structured version Visualization version GIF version |
Description: 𝐺 (see om2uz0i 12608) 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 12614 | . 2 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶) |
4 | epel 4952 | . . . 4 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
5 | 1, 2 | om2uzlt2i 12612 | . . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 ∈ 𝑧 ↔ (𝐺‘𝑦) < (𝐺‘𝑧))) |
6 | 4, 5 | syl5bb 271 | . . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 E 𝑧 ↔ (𝐺‘𝑦) < (𝐺‘𝑧))) |
7 | 6 | rgen2a 2960 | . 2 ⊢ ∀𝑦 ∈ ω ∀𝑧 ∈ ω (𝑦 E 𝑧 ↔ (𝐺‘𝑦) < (𝐺‘𝑧)) |
8 | df-isom 5813 | . 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 957 | 1 ⊢ 𝐺 Isom E , < (ω, (ℤ≥‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 E cep 4947 ↾ cres 5040 –1-1-onto→wf1o 5803 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 ωcom 6957 reccrdg 7392 1c1 9816 + caddc 9818 < clt 9953 ℤcz 11254 ℤ≥cuz 11563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 |
This theorem is referenced by: om2uzoi 12616 ltweuz 12622 fz1isolem 13102 |
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