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Mathbox for Jim Kingdon |
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Mirrors > Home > ILE Home > Th. List > Mathboxes > 2o01f | GIF version |
Description: Mapping zero and one between ω and ℕ0 style integers. (Contributed by Jim Kingdon, 28-Jun-2024.) |
Ref | Expression |
---|---|
012of.g | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) |
Ref | Expression |
---|---|
2o01f | ⊢ (𝐺 ↾ 2o):2o⟶{0, 1} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 012of.g | . . . . . 6 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
2 | 1 | frechashgf1o 10305 | . . . . 5 ⊢ 𝐺:ω–1-1-onto→ℕ0 |
3 | f1of 5407 | . . . . 5 ⊢ (𝐺:ω–1-1-onto→ℕ0 → 𝐺:ω⟶ℕ0) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 𝐺:ω⟶ℕ0 |
5 | 2onn 6457 | . . . . 5 ⊢ 2o ∈ ω | |
6 | omelon 4562 | . . . . . 6 ⊢ ω ∈ On | |
7 | 6 | onelssi 4384 | . . . . 5 ⊢ (2o ∈ ω → 2o ⊆ ω) |
8 | 5, 7 | ax-mp 5 | . . . 4 ⊢ 2o ⊆ ω |
9 | fssres 5338 | . . . 4 ⊢ ((𝐺:ω⟶ℕ0 ∧ 2o ⊆ ω) → (𝐺 ↾ 2o):2o⟶ℕ0) | |
10 | 4, 8, 9 | mp2an 423 | . . 3 ⊢ (𝐺 ↾ 2o):2o⟶ℕ0 |
11 | ffn 5312 | . . 3 ⊢ ((𝐺 ↾ 2o):2o⟶ℕ0 → (𝐺 ↾ 2o) Fn 2o) | |
12 | 10, 11 | ax-mp 5 | . 2 ⊢ (𝐺 ↾ 2o) Fn 2o |
13 | fvres 5485 | . . . 4 ⊢ (𝑗 ∈ 2o → ((𝐺 ↾ 2o)‘𝑗) = (𝐺‘𝑗)) | |
14 | elpri 3579 | . . . . . 6 ⊢ (𝑗 ∈ {∅, 1o} → (𝑗 = ∅ ∨ 𝑗 = 1o)) | |
15 | df2o3 6367 | . . . . . 6 ⊢ 2o = {∅, 1o} | |
16 | 14, 15 | eleq2s 2249 | . . . . 5 ⊢ (𝑗 ∈ 2o → (𝑗 = ∅ ∨ 𝑗 = 1o)) |
17 | fveq2 5461 | . . . . . . 7 ⊢ (𝑗 = ∅ → (𝐺‘𝑗) = (𝐺‘∅)) | |
18 | 0zd 9158 | . . . . . . . . . 10 ⊢ (⊤ → 0 ∈ ℤ) | |
19 | 18, 1 | frec2uz0d 10276 | . . . . . . . . 9 ⊢ (⊤ → (𝐺‘∅) = 0) |
20 | 19 | mptru 1341 | . . . . . . . 8 ⊢ (𝐺‘∅) = 0 |
21 | c0ex 7851 | . . . . . . . . 9 ⊢ 0 ∈ V | |
22 | 21 | prid1 3661 | . . . . . . . 8 ⊢ 0 ∈ {0, 1} |
23 | 20, 22 | eqeltri 2227 | . . . . . . 7 ⊢ (𝐺‘∅) ∈ {0, 1} |
24 | 17, 23 | eqeltrdi 2245 | . . . . . 6 ⊢ (𝑗 = ∅ → (𝐺‘𝑗) ∈ {0, 1}) |
25 | fveq2 5461 | . . . . . . 7 ⊢ (𝑗 = 1o → (𝐺‘𝑗) = (𝐺‘1o)) | |
26 | df-1o 6353 | . . . . . . . . . 10 ⊢ 1o = suc ∅ | |
27 | 26 | fveq2i 5464 | . . . . . . . . 9 ⊢ (𝐺‘1o) = (𝐺‘suc ∅) |
28 | peano1 4547 | . . . . . . . . . . . 12 ⊢ ∅ ∈ ω | |
29 | 28 | a1i 9 | . . . . . . . . . . 11 ⊢ (⊤ → ∅ ∈ ω) |
30 | 18, 1, 29 | frec2uzsucd 10278 | . . . . . . . . . 10 ⊢ (⊤ → (𝐺‘suc ∅) = ((𝐺‘∅) + 1)) |
31 | 30 | mptru 1341 | . . . . . . . . 9 ⊢ (𝐺‘suc ∅) = ((𝐺‘∅) + 1) |
32 | 20 | oveq1i 5824 | . . . . . . . . . 10 ⊢ ((𝐺‘∅) + 1) = (0 + 1) |
33 | 0p1e1 8926 | . . . . . . . . . 10 ⊢ (0 + 1) = 1 | |
34 | 32, 33 | eqtri 2175 | . . . . . . . . 9 ⊢ ((𝐺‘∅) + 1) = 1 |
35 | 27, 31, 34 | 3eqtri 2179 | . . . . . . . 8 ⊢ (𝐺‘1o) = 1 |
36 | 1ex 7852 | . . . . . . . . 9 ⊢ 1 ∈ V | |
37 | 36 | prid2 3662 | . . . . . . . 8 ⊢ 1 ∈ {0, 1} |
38 | 35, 37 | eqeltri 2227 | . . . . . . 7 ⊢ (𝐺‘1o) ∈ {0, 1} |
39 | 25, 38 | eqeltrdi 2245 | . . . . . 6 ⊢ (𝑗 = 1o → (𝐺‘𝑗) ∈ {0, 1}) |
40 | 24, 39 | jaoi 706 | . . . . 5 ⊢ ((𝑗 = ∅ ∨ 𝑗 = 1o) → (𝐺‘𝑗) ∈ {0, 1}) |
41 | 16, 40 | syl 14 | . . . 4 ⊢ (𝑗 ∈ 2o → (𝐺‘𝑗) ∈ {0, 1}) |
42 | 13, 41 | eqeltrd 2231 | . . 3 ⊢ (𝑗 ∈ 2o → ((𝐺 ↾ 2o)‘𝑗) ∈ {0, 1}) |
43 | 42 | rgen 2507 | . 2 ⊢ ∀𝑗 ∈ 2o ((𝐺 ↾ 2o)‘𝑗) ∈ {0, 1} |
44 | ffnfv 5618 | . 2 ⊢ ((𝐺 ↾ 2o):2o⟶{0, 1} ↔ ((𝐺 ↾ 2o) Fn 2o ∧ ∀𝑗 ∈ 2o ((𝐺 ↾ 2o)‘𝑗) ∈ {0, 1})) | |
45 | 12, 43, 44 | mpbir2an 927 | 1 ⊢ (𝐺 ↾ 2o):2o⟶{0, 1} |
Colors of variables: wff set class |
Syntax hints: ∨ wo 698 = wceq 1332 ⊤wtru 1333 ∈ wcel 2125 ∀wral 2432 ⊆ wss 3098 ∅c0 3390 {cpr 3557 ↦ cmpt 4021 suc csuc 4320 ωcom 4543 ↾ cres 4581 Fn wfn 5158 ⟶wf 5159 –1-1-onto→wf1o 5162 ‘cfv 5163 (class class class)co 5814 freccfrec 6327 1oc1o 6346 2oc2o 6347 0cc0 7711 1c1 7712 + caddc 7714 ℕ0cn0 9069 ℤcz 9146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-addcom 7811 ax-addass 7813 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-0id 7819 ax-rnegex 7820 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-ltadd 7827 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-recs 6242 df-frec 6328 df-1o 6353 df-2o 6354 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-inn 8813 df-n0 9070 df-z 9147 df-uz 9419 |
This theorem is referenced by: isomninnlem 13542 iswomninnlem 13561 ismkvnnlem 13564 |
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