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| Mirrors > Home > MPE Home > Th. List > om2uzf1oi | Structured version Visualization version GIF version | ||
| Description: 𝐺 (see om2uz0i 13988) 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 8475 | . . . . 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 6665 | . . . . 5 ⊢ (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) Fn ω) | 
| 4 | 1, 3 | mpbir 231 | . . . 4 ⊢ 𝐺 Fn ω | 
| 5 | om2uz.1 | . . . . . 6 ⊢ 𝐶 ∈ ℤ | |
| 6 | 5, 2 | om2uzrani 13993 | . . . . 5 ⊢ ran 𝐺 = (ℤ≥‘𝐶) | 
| 7 | 6 | eqimssi 4044 | . . . 4 ⊢ ran 𝐺 ⊆ (ℤ≥‘𝐶) | 
| 8 | df-f 6565 | . . . 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 13990 | . . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ (ℤ≥‘𝐶)) | 
| 11 | eluzelz 12888 | . . . . . . . . 9 ⊢ ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) → (𝐺‘𝑦) ∈ ℤ) | |
| 12 | 10, 11 | syl 17 | . . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℤ) | 
| 13 | 12 | zred 12722 | . . . . . . 7 ⊢ (𝑦 ∈ ω → (𝐺‘𝑦) ∈ ℝ) | 
| 14 | 5, 2 | om2uzuzi 13990 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ (ℤ≥‘𝐶)) | 
| 15 | eluzelz 12888 | . . . . . . . . 9 ⊢ ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) → (𝐺‘𝑧) ∈ ℤ) | |
| 16 | 14, 15 | syl 17 | . . . . . . . 8 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℤ) | 
| 17 | 16 | zred 12722 | . . . . . . 7 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℝ) | 
| 18 | lttri3 11344 | . . . . . . 7 ⊢ (((𝐺‘𝑦) ∈ ℝ ∧ (𝐺‘𝑧) ∈ ℝ) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ (¬ (𝐺‘𝑦) < (𝐺‘𝑧) ∧ ¬ (𝐺‘𝑧) < (𝐺‘𝑦)))) | |
| 19 | 13, 17, 18 | syl2an 596 | . . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ (¬ (𝐺‘𝑦) < (𝐺‘𝑧) ∧ ¬ (𝐺‘𝑧) < (𝐺‘𝑦)))) | 
| 20 | ioran 986 | . . . . . 6 ⊢ (¬ ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)) ↔ (¬ (𝐺‘𝑦) < (𝐺‘𝑧) ∧ ¬ (𝐺‘𝑧) < (𝐺‘𝑦))) | |
| 21 | 19, 20 | bitr4di 289 | . . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ ¬ ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)))) | 
| 22 | nnord 7895 | . . . . . . . . 9 ⊢ (𝑦 ∈ ω → Ord 𝑦) | |
| 23 | nnord 7895 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → Ord 𝑧) | |
| 24 | ordtri3 6420 | . . . . . . . . 9 ⊢ ((Ord 𝑦 ∧ Ord 𝑧) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))) | |
| 25 | 22, 23, 24 | syl2an 596 | . . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))) | 
| 26 | 25 | con2bid 354 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦) ↔ ¬ 𝑦 = 𝑧)) | 
| 27 | 5, 2 | om2uzlti 13991 | . . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 ∈ 𝑧 → (𝐺‘𝑦) < (𝐺‘𝑧))) | 
| 28 | 5, 2 | om2uzlti 13991 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ω ∧ 𝑦 ∈ ω) → (𝑧 ∈ 𝑦 → (𝐺‘𝑧) < (𝐺‘𝑦))) | 
| 29 | 28 | ancoms 458 | . . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑧 ∈ 𝑦 → (𝐺‘𝑧) < (𝐺‘𝑦))) | 
| 30 | 27, 29 | orim12d 967 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦) → ((𝐺‘𝑦) < (𝐺‘𝑧) ∨ (𝐺‘𝑧) < (𝐺‘𝑦)))) | 
| 31 | 26, 30 | sylbird 260 | . . . . . 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 240 | . . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)) | 
| 34 | 33 | rgen2 3199 | . . 3 ⊢ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧) | 
| 35 | dff13 7275 | . . 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 6857 | . 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 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ↾ cres 5687 Ord word 6383 Fn wfn 6556 ⟶wf 6557 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ωcom 7887 reccrdg 8449 ℝcr 11154 1c1 11156 + caddc 11158 < clt 11295 ℤcz 12613 ℤ≥cuz 12878 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 | 
| This theorem is referenced by: om2uzisoi 13995 uzrdglem 13998 uzrdgfni 13999 uzrdgsuci 14001 uzenom 14005 fzennn 14009 cardfz 14011 hashgf1o 14012 axdc4uzlem 14024 unbenlem 16946 | 
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