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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 10401 | . . . . 5 ⊢ 𝐺:ω–1-1-onto→ℕ0 |
3 | f1of 5456 | . . . . 5 ⊢ (𝐺:ω–1-1-onto→ℕ0 → 𝐺:ω⟶ℕ0) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 𝐺:ω⟶ℕ0 |
5 | 2onn 6515 | . . . . 5 ⊢ 2o ∈ ω | |
6 | omelon 4604 | . . . . . 6 ⊢ ω ∈ On | |
7 | 6 | onelssi 4425 | . . . . 5 ⊢ (2o ∈ ω → 2o ⊆ ω) |
8 | 5, 7 | ax-mp 5 | . . . 4 ⊢ 2o ⊆ ω |
9 | fssres 5386 | . . . 4 ⊢ ((𝐺:ω⟶ℕ0 ∧ 2o ⊆ ω) → (𝐺 ↾ 2o):2o⟶ℕ0) | |
10 | 4, 8, 9 | mp2an 426 | . . 3 ⊢ (𝐺 ↾ 2o):2o⟶ℕ0 |
11 | ffn 5360 | . . 3 ⊢ ((𝐺 ↾ 2o):2o⟶ℕ0 → (𝐺 ↾ 2o) Fn 2o) | |
12 | 10, 11 | ax-mp 5 | . 2 ⊢ (𝐺 ↾ 2o) Fn 2o |
13 | fvres 5534 | . . . 4 ⊢ (𝑗 ∈ 2o → ((𝐺 ↾ 2o)‘𝑗) = (𝐺‘𝑗)) | |
14 | elpri 3614 | . . . . . 6 ⊢ (𝑗 ∈ {∅, 1o} → (𝑗 = ∅ ∨ 𝑗 = 1o)) | |
15 | df2o3 6424 | . . . . . 6 ⊢ 2o = {∅, 1o} | |
16 | 14, 15 | eleq2s 2272 | . . . . 5 ⊢ (𝑗 ∈ 2o → (𝑗 = ∅ ∨ 𝑗 = 1o)) |
17 | fveq2 5510 | . . . . . . 7 ⊢ (𝑗 = ∅ → (𝐺‘𝑗) = (𝐺‘∅)) | |
18 | 0zd 9241 | . . . . . . . . . 10 ⊢ (⊤ → 0 ∈ ℤ) | |
19 | 18, 1 | frec2uz0d 10372 | . . . . . . . . 9 ⊢ (⊤ → (𝐺‘∅) = 0) |
20 | 19 | mptru 1362 | . . . . . . . 8 ⊢ (𝐺‘∅) = 0 |
21 | c0ex 7929 | . . . . . . . . 9 ⊢ 0 ∈ V | |
22 | 21 | prid1 3697 | . . . . . . . 8 ⊢ 0 ∈ {0, 1} |
23 | 20, 22 | eqeltri 2250 | . . . . . . 7 ⊢ (𝐺‘∅) ∈ {0, 1} |
24 | 17, 23 | eqeltrdi 2268 | . . . . . 6 ⊢ (𝑗 = ∅ → (𝐺‘𝑗) ∈ {0, 1}) |
25 | fveq2 5510 | . . . . . . 7 ⊢ (𝑗 = 1o → (𝐺‘𝑗) = (𝐺‘1o)) | |
26 | df-1o 6410 | . . . . . . . . . 10 ⊢ 1o = suc ∅ | |
27 | 26 | fveq2i 5513 | . . . . . . . . 9 ⊢ (𝐺‘1o) = (𝐺‘suc ∅) |
28 | peano1 4589 | . . . . . . . . . . . 12 ⊢ ∅ ∈ ω | |
29 | 28 | a1i 9 | . . . . . . . . . . 11 ⊢ (⊤ → ∅ ∈ ω) |
30 | 18, 1, 29 | frec2uzsucd 10374 | . . . . . . . . . 10 ⊢ (⊤ → (𝐺‘suc ∅) = ((𝐺‘∅) + 1)) |
31 | 30 | mptru 1362 | . . . . . . . . 9 ⊢ (𝐺‘suc ∅) = ((𝐺‘∅) + 1) |
32 | 20 | oveq1i 5878 | . . . . . . . . . 10 ⊢ ((𝐺‘∅) + 1) = (0 + 1) |
33 | 0p1e1 9009 | . . . . . . . . . 10 ⊢ (0 + 1) = 1 | |
34 | 32, 33 | eqtri 2198 | . . . . . . . . 9 ⊢ ((𝐺‘∅) + 1) = 1 |
35 | 27, 31, 34 | 3eqtri 2202 | . . . . . . . 8 ⊢ (𝐺‘1o) = 1 |
36 | 1ex 7930 | . . . . . . . . 9 ⊢ 1 ∈ V | |
37 | 36 | prid2 3698 | . . . . . . . 8 ⊢ 1 ∈ {0, 1} |
38 | 35, 37 | eqeltri 2250 | . . . . . . 7 ⊢ (𝐺‘1o) ∈ {0, 1} |
39 | 25, 38 | eqeltrdi 2268 | . . . . . 6 ⊢ (𝑗 = 1o → (𝐺‘𝑗) ∈ {0, 1}) |
40 | 24, 39 | jaoi 716 | . . . . 5 ⊢ ((𝑗 = ∅ ∨ 𝑗 = 1o) → (𝐺‘𝑗) ∈ {0, 1}) |
41 | 16, 40 | syl 14 | . . . 4 ⊢ (𝑗 ∈ 2o → (𝐺‘𝑗) ∈ {0, 1}) |
42 | 13, 41 | eqeltrd 2254 | . . 3 ⊢ (𝑗 ∈ 2o → ((𝐺 ↾ 2o)‘𝑗) ∈ {0, 1}) |
43 | 42 | rgen 2530 | . 2 ⊢ ∀𝑗 ∈ 2o ((𝐺 ↾ 2o)‘𝑗) ∈ {0, 1} |
44 | ffnfv 5669 | . 2 ⊢ ((𝐺 ↾ 2o):2o⟶{0, 1} ↔ ((𝐺 ↾ 2o) Fn 2o ∧ ∀𝑗 ∈ 2o ((𝐺 ↾ 2o)‘𝑗) ∈ {0, 1})) | |
45 | 12, 43, 44 | mpbir2an 942 | 1 ⊢ (𝐺 ↾ 2o):2o⟶{0, 1} |
Colors of variables: wff set class |
Syntax hints: ∨ wo 708 = wceq 1353 ⊤wtru 1354 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3129 ∅c0 3422 {cpr 3592 ↦ cmpt 4061 suc csuc 4361 ωcom 4585 ↾ cres 4624 Fn wfn 5206 ⟶wf 5207 –1-1-onto→wf1o 5210 ‘cfv 5211 (class class class)co 5868 freccfrec 6384 1oc1o 6403 2oc2o 6404 0cc0 7789 1c1 7790 + caddc 7792 ℕ0cn0 9152 ℤcz 9229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-recs 6299 df-frec 6385 df-1o 6410 df-2o 6411 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-n0 9153 df-z 9230 df-uz 9505 |
This theorem is referenced by: isomninnlem 14401 iswomninnlem 14420 ismkvnnlem 14423 |
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