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Mirrors > Home > ILE Home > Th. List > frechashgf1o | GIF version |
Description: 𝐺 maps ω one-to-one onto ℕ0. (Contributed by Jim Kingdon, 19-May-2020.) |
Ref | Expression |
---|---|
frecfzennn.1 | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) |
Ref | Expression |
---|---|
frechashgf1o | ⊢ 𝐺:ω–1-1-onto→ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0zd 9263 | . . . 4 ⊢ (⊤ → 0 ∈ ℤ) | |
2 | frecfzennn.1 | . . . 4 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
3 | 1, 2 | frec2uzf1od 10403 | . . 3 ⊢ (⊤ → 𝐺:ω–1-1-onto→(ℤ≥‘0)) |
4 | 3 | mptru 1362 | . 2 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘0) |
5 | nn0uz 9560 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
6 | f1oeq3 5451 | . . 3 ⊢ (ℕ0 = (ℤ≥‘0) → (𝐺:ω–1-1-onto→ℕ0 ↔ 𝐺:ω–1-1-onto→(ℤ≥‘0))) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ (𝐺:ω–1-1-onto→ℕ0 ↔ 𝐺:ω–1-1-onto→(ℤ≥‘0)) |
8 | 4, 7 | mpbir 146 | 1 ⊢ 𝐺:ω–1-1-onto→ℕ0 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 ⊤wtru 1354 ↦ cmpt 4064 ωcom 4589 –1-1-onto→wf1o 5215 ‘cfv 5216 (class class class)co 5874 freccfrec 6390 0cc0 7810 1c1 7811 + caddc 7813 ℕ0cn0 9174 ℤcz 9251 ℤ≥cuz 9526 |
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 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-recs 6305 df-frec 6391 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-inn 8918 df-n0 9175 df-z 9252 df-uz 9527 |
This theorem is referenced by: fzfig 10427 nnenom 10431 fnn0nninf 10434 0tonninf 10436 1tonninf 10437 omgadd 10777 ennnfonelemp1 12401 ennnfonelemhdmp1 12404 ennnfonelemss 12405 ennnfonelemkh 12407 ennnfonelemhf1o 12408 ennnfonelemex 12409 ennnfonelemnn0 12417 ctinfomlemom 12422 012of 14627 2o01f 14628 isomninnlem 14660 iswomninnlem 14679 ismkvnnlem 14682 |
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