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 10353 | . . . . 5 ⊢ 𝐺:ω–1-1-onto→ℕ0 |
3 | f1of 5426 | . . . . 5 ⊢ (𝐺:ω–1-1-onto→ℕ0 → 𝐺:ω⟶ℕ0) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 𝐺:ω⟶ℕ0 |
5 | 2onn 6480 | . . . . 5 ⊢ 2o ∈ ω | |
6 | omelon 4580 | . . . . . 6 ⊢ ω ∈ On | |
7 | 6 | onelssi 4401 | . . . . 5 ⊢ (2o ∈ ω → 2o ⊆ ω) |
8 | 5, 7 | ax-mp 5 | . . . 4 ⊢ 2o ⊆ ω |
9 | fssres 5357 | . . . 4 ⊢ ((𝐺:ω⟶ℕ0 ∧ 2o ⊆ ω) → (𝐺 ↾ 2o):2o⟶ℕ0) | |
10 | 4, 8, 9 | mp2an 423 | . . 3 ⊢ (𝐺 ↾ 2o):2o⟶ℕ0 |
11 | ffn 5331 | . . 3 ⊢ ((𝐺 ↾ 2o):2o⟶ℕ0 → (𝐺 ↾ 2o) Fn 2o) | |
12 | 10, 11 | ax-mp 5 | . 2 ⊢ (𝐺 ↾ 2o) Fn 2o |
13 | fvres 5504 | . . . 4 ⊢ (𝑗 ∈ 2o → ((𝐺 ↾ 2o)‘𝑗) = (𝐺‘𝑗)) | |
14 | elpri 3593 | . . . . . 6 ⊢ (𝑗 ∈ {∅, 1o} → (𝑗 = ∅ ∨ 𝑗 = 1o)) | |
15 | df2o3 6389 | . . . . . 6 ⊢ 2o = {∅, 1o} | |
16 | 14, 15 | eleq2s 2259 | . . . . 5 ⊢ (𝑗 ∈ 2o → (𝑗 = ∅ ∨ 𝑗 = 1o)) |
17 | fveq2 5480 | . . . . . . 7 ⊢ (𝑗 = ∅ → (𝐺‘𝑗) = (𝐺‘∅)) | |
18 | 0zd 9194 | . . . . . . . . . 10 ⊢ (⊤ → 0 ∈ ℤ) | |
19 | 18, 1 | frec2uz0d 10324 | . . . . . . . . 9 ⊢ (⊤ → (𝐺‘∅) = 0) |
20 | 19 | mptru 1351 | . . . . . . . 8 ⊢ (𝐺‘∅) = 0 |
21 | c0ex 7884 | . . . . . . . . 9 ⊢ 0 ∈ V | |
22 | 21 | prid1 3676 | . . . . . . . 8 ⊢ 0 ∈ {0, 1} |
23 | 20, 22 | eqeltri 2237 | . . . . . . 7 ⊢ (𝐺‘∅) ∈ {0, 1} |
24 | 17, 23 | eqeltrdi 2255 | . . . . . 6 ⊢ (𝑗 = ∅ → (𝐺‘𝑗) ∈ {0, 1}) |
25 | fveq2 5480 | . . . . . . 7 ⊢ (𝑗 = 1o → (𝐺‘𝑗) = (𝐺‘1o)) | |
26 | df-1o 6375 | . . . . . . . . . 10 ⊢ 1o = suc ∅ | |
27 | 26 | fveq2i 5483 | . . . . . . . . 9 ⊢ (𝐺‘1o) = (𝐺‘suc ∅) |
28 | peano1 4565 | . . . . . . . . . . . 12 ⊢ ∅ ∈ ω | |
29 | 28 | a1i 9 | . . . . . . . . . . 11 ⊢ (⊤ → ∅ ∈ ω) |
30 | 18, 1, 29 | frec2uzsucd 10326 | . . . . . . . . . 10 ⊢ (⊤ → (𝐺‘suc ∅) = ((𝐺‘∅) + 1)) |
31 | 30 | mptru 1351 | . . . . . . . . 9 ⊢ (𝐺‘suc ∅) = ((𝐺‘∅) + 1) |
32 | 20 | oveq1i 5846 | . . . . . . . . . 10 ⊢ ((𝐺‘∅) + 1) = (0 + 1) |
33 | 0p1e1 8962 | . . . . . . . . . 10 ⊢ (0 + 1) = 1 | |
34 | 32, 33 | eqtri 2185 | . . . . . . . . 9 ⊢ ((𝐺‘∅) + 1) = 1 |
35 | 27, 31, 34 | 3eqtri 2189 | . . . . . . . 8 ⊢ (𝐺‘1o) = 1 |
36 | 1ex 7885 | . . . . . . . . 9 ⊢ 1 ∈ V | |
37 | 36 | prid2 3677 | . . . . . . . 8 ⊢ 1 ∈ {0, 1} |
38 | 35, 37 | eqeltri 2237 | . . . . . . 7 ⊢ (𝐺‘1o) ∈ {0, 1} |
39 | 25, 38 | eqeltrdi 2255 | . . . . . 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 2241 | . . 3 ⊢ (𝑗 ∈ 2o → ((𝐺 ↾ 2o)‘𝑗) ∈ {0, 1}) |
43 | 42 | rgen 2517 | . 2 ⊢ ∀𝑗 ∈ 2o ((𝐺 ↾ 2o)‘𝑗) ∈ {0, 1} |
44 | ffnfv 5637 | . 2 ⊢ ((𝐺 ↾ 2o):2o⟶{0, 1} ↔ ((𝐺 ↾ 2o) Fn 2o ∧ ∀𝑗 ∈ 2o ((𝐺 ↾ 2o)‘𝑗) ∈ {0, 1})) | |
45 | 12, 43, 44 | mpbir2an 931 | 1 ⊢ (𝐺 ↾ 2o):2o⟶{0, 1} |
Colors of variables: wff set class |
Syntax hints: ∨ wo 698 = wceq 1342 ⊤wtru 1343 ∈ wcel 2135 ∀wral 2442 ⊆ wss 3111 ∅c0 3404 {cpr 3571 ↦ cmpt 4037 suc csuc 4337 ωcom 4561 ↾ cres 4600 Fn wfn 5177 ⟶wf 5178 –1-1-onto→wf1o 5181 ‘cfv 5182 (class class class)co 5836 freccfrec 6349 1oc1o 6368 2oc2o 6369 0cc0 7744 1c1 7745 + caddc 7747 ℕ0cn0 9105 ℤcz 9182 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-recs 6264 df-frec 6350 df-1o 6375 df-2o 6376 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 |
This theorem is referenced by: isomninnlem 13750 iswomninnlem 13769 ismkvnnlem 13772 |
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