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Mirrors > Home > MPE Home > Th. List > hashpw | Structured version Visualization version GIF version |
Description: The size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.) (Proof shortened by Mario Carneiro, 5-Aug-2014.) |
Ref | Expression |
---|---|
hashpw | ⊢ (𝐴 ∈ Fin → (#‘𝒫 𝐴) = (2↑(#‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4194 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | fveq2d 6233 | . . 3 ⊢ (𝑥 = 𝐴 → (#‘𝒫 𝑥) = (#‘𝒫 𝐴)) |
3 | fveq2 6229 | . . . 4 ⊢ (𝑥 = 𝐴 → (#‘𝑥) = (#‘𝐴)) | |
4 | 3 | oveq2d 6706 | . . 3 ⊢ (𝑥 = 𝐴 → (2↑(#‘𝑥)) = (2↑(#‘𝐴))) |
5 | 2, 4 | eqeq12d 2666 | . 2 ⊢ (𝑥 = 𝐴 → ((#‘𝒫 𝑥) = (2↑(#‘𝑥)) ↔ (#‘𝒫 𝐴) = (2↑(#‘𝐴)))) |
6 | vex 3234 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | 6 | pw2en 8108 | . . . 4 ⊢ 𝒫 𝑥 ≈ (2𝑜 ↑𝑚 𝑥) |
8 | pwfi 8302 | . . . . . 6 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin) | |
9 | 8 | biimpi 206 | . . . . 5 ⊢ (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin) |
10 | df2o2 7619 | . . . . . . 7 ⊢ 2𝑜 = {∅, {∅}} | |
11 | prfi 8276 | . . . . . . 7 ⊢ {∅, {∅}} ∈ Fin | |
12 | 10, 11 | eqeltri 2726 | . . . . . 6 ⊢ 2𝑜 ∈ Fin |
13 | mapfi 8303 | . . . . . 6 ⊢ ((2𝑜 ∈ Fin ∧ 𝑥 ∈ Fin) → (2𝑜 ↑𝑚 𝑥) ∈ Fin) | |
14 | 12, 13 | mpan 706 | . . . . 5 ⊢ (𝑥 ∈ Fin → (2𝑜 ↑𝑚 𝑥) ∈ Fin) |
15 | hashen 13175 | . . . . 5 ⊢ ((𝒫 𝑥 ∈ Fin ∧ (2𝑜 ↑𝑚 𝑥) ∈ Fin) → ((#‘𝒫 𝑥) = (#‘(2𝑜 ↑𝑚 𝑥)) ↔ 𝒫 𝑥 ≈ (2𝑜 ↑𝑚 𝑥))) | |
16 | 9, 14, 15 | syl2anc 694 | . . . 4 ⊢ (𝑥 ∈ Fin → ((#‘𝒫 𝑥) = (#‘(2𝑜 ↑𝑚 𝑥)) ↔ 𝒫 𝑥 ≈ (2𝑜 ↑𝑚 𝑥))) |
17 | 7, 16 | mpbiri 248 | . . 3 ⊢ (𝑥 ∈ Fin → (#‘𝒫 𝑥) = (#‘(2𝑜 ↑𝑚 𝑥))) |
18 | hashmap 13260 | . . . . 5 ⊢ ((2𝑜 ∈ Fin ∧ 𝑥 ∈ Fin) → (#‘(2𝑜 ↑𝑚 𝑥)) = ((#‘2𝑜)↑(#‘𝑥))) | |
19 | 12, 18 | mpan 706 | . . . 4 ⊢ (𝑥 ∈ Fin → (#‘(2𝑜 ↑𝑚 𝑥)) = ((#‘2𝑜)↑(#‘𝑥))) |
20 | hash2 13231 | . . . . 5 ⊢ (#‘2𝑜) = 2 | |
21 | 20 | oveq1i 6700 | . . . 4 ⊢ ((#‘2𝑜)↑(#‘𝑥)) = (2↑(#‘𝑥)) |
22 | 19, 21 | syl6eq 2701 | . . 3 ⊢ (𝑥 ∈ Fin → (#‘(2𝑜 ↑𝑚 𝑥)) = (2↑(#‘𝑥))) |
23 | 17, 22 | eqtrd 2685 | . 2 ⊢ (𝑥 ∈ Fin → (#‘𝒫 𝑥) = (2↑(#‘𝑥))) |
24 | 5, 23 | vtoclga 3303 | 1 ⊢ (𝐴 ∈ Fin → (#‘𝒫 𝐴) = (2↑(#‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1523 ∈ wcel 2030 ∅c0 3948 𝒫 cpw 4191 {csn 4210 {cpr 4212 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 2𝑜c2o 7599 ↑𝑚 cmap 7899 ≈ cen 7994 Fincfn 7997 2c2 11108 ↑cexp 12900 #chash 13157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-seq 12842 df-exp 12901 df-hash 13158 |
This theorem is referenced by: ackbijnn 14604 |
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