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Mirrors > Home > MPE Home > Th. List > cardfz | Structured version Visualization version GIF version |
Description: The cardinality of a finite set of sequential integers. (See om2uz0i 13314 for a description of the hypothesis.) (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
fzennn.1 | ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) |
Ref | Expression |
---|---|
cardfz | ⊢ (𝑁 ∈ ℕ0 → (card‘(1...𝑁)) = (◡𝐺‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzennn.1 | . . . 4 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | |
2 | 1 | fzennn 13335 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡𝐺‘𝑁)) |
3 | carden2b 9395 | . . 3 ⊢ ((1...𝑁) ≈ (◡𝐺‘𝑁) → (card‘(1...𝑁)) = (card‘(◡𝐺‘𝑁))) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → (card‘(1...𝑁)) = (card‘(◡𝐺‘𝑁))) |
5 | 0z 11991 | . . . . 5 ⊢ 0 ∈ ℤ | |
6 | 5, 1 | om2uzf1oi 13320 | . . . 4 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘0) |
7 | elnn0uz 12282 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
8 | 7 | biimpi 218 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0)) |
9 | f1ocnvdm 7040 | . . . 4 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘0)) → (◡𝐺‘𝑁) ∈ ω) | |
10 | 6, 8, 9 | sylancr 589 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (◡𝐺‘𝑁) ∈ ω) |
11 | cardnn 9391 | . . 3 ⊢ ((◡𝐺‘𝑁) ∈ ω → (card‘(◡𝐺‘𝑁)) = (◡𝐺‘𝑁)) | |
12 | 10, 11 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → (card‘(◡𝐺‘𝑁)) = (◡𝐺‘𝑁)) |
13 | 4, 12 | eqtrd 2856 | 1 ⊢ (𝑁 ∈ ℕ0 → (card‘(1...𝑁)) = (◡𝐺‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5065 ↦ cmpt 5145 ◡ccnv 5553 ↾ cres 5556 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 ωcom 7579 reccrdg 8044 ≈ cen 8505 cardccrd 9363 0cc0 10536 1c1 10537 + caddc 10539 ℕ0cn0 11896 ℤ≥cuz 12242 ...cfz 12891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 |
This theorem is referenced by: hashfz1 13705 |
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