Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 13665 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 13686 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡𝐺‘𝑁)) |
3 | carden2b 9726 | . . 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 12330 | . . . . 5 ⊢ 0 ∈ ℤ | |
6 | 5, 1 | om2uzf1oi 13671 | . . . 4 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘0) |
7 | elnn0uz 12622 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
8 | 7 | biimpi 215 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0)) |
9 | f1ocnvdm 7153 | . . . 4 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘0)) → (◡𝐺‘𝑁) ∈ ω) | |
10 | 6, 8, 9 | sylancr 587 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (◡𝐺‘𝑁) ∈ ω) |
11 | cardnn 9722 | . . 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 2780 | 1 ⊢ (𝑁 ∈ ℕ0 → (card‘(1...𝑁)) = (◡𝐺‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 Vcvv 3431 class class class wbr 5079 ↦ cmpt 5162 ◡ccnv 5589 ↾ cres 5592 –1-1-onto→wf1o 6431 ‘cfv 6432 (class class class)co 7271 ωcom 7706 reccrdg 8231 ≈ cen 8713 cardccrd 9694 0cc0 10872 1c1 10873 + caddc 10875 ℕ0cn0 12233 ℤ≥cuz 12581 ...cfz 13238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 |
This theorem is referenced by: hashfz1 14058 |
Copyright terms: Public domain | W3C validator |