![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4617 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | fveq2d 6896 | . . 3 ⊢ (𝑥 = 𝐴 → (♯‘𝒫 𝑥) = (♯‘𝒫 𝐴)) |
3 | fveq2 6892 | . . . 4 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴)) | |
4 | 3 | oveq2d 7425 | . . 3 ⊢ (𝑥 = 𝐴 → (2↑(♯‘𝑥)) = (2↑(♯‘𝐴))) |
5 | 2, 4 | eqeq12d 2749 | . 2 ⊢ (𝑥 = 𝐴 → ((♯‘𝒫 𝑥) = (2↑(♯‘𝑥)) ↔ (♯‘𝒫 𝐴) = (2↑(♯‘𝐴)))) |
6 | vex 3479 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | 6 | pw2en 9079 | . . . 4 ⊢ 𝒫 𝑥 ≈ (2o ↑m 𝑥) |
8 | pwfi 9178 | . . . . . 6 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin) | |
9 | 8 | biimpi 215 | . . . . 5 ⊢ (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin) |
10 | df2o2 8475 | . . . . . . 7 ⊢ 2o = {∅, {∅}} | |
11 | prfi 9322 | . . . . . . 7 ⊢ {∅, {∅}} ∈ Fin | |
12 | 10, 11 | eqeltri 2830 | . . . . . 6 ⊢ 2o ∈ Fin |
13 | mapfi 9348 | . . . . . 6 ⊢ ((2o ∈ Fin ∧ 𝑥 ∈ Fin) → (2o ↑m 𝑥) ∈ Fin) | |
14 | 12, 13 | mpan 689 | . . . . 5 ⊢ (𝑥 ∈ Fin → (2o ↑m 𝑥) ∈ Fin) |
15 | hashen 14307 | . . . . 5 ⊢ ((𝒫 𝑥 ∈ Fin ∧ (2o ↑m 𝑥) ∈ Fin) → ((♯‘𝒫 𝑥) = (♯‘(2o ↑m 𝑥)) ↔ 𝒫 𝑥 ≈ (2o ↑m 𝑥))) | |
16 | 9, 14, 15 | syl2anc 585 | . . . 4 ⊢ (𝑥 ∈ Fin → ((♯‘𝒫 𝑥) = (♯‘(2o ↑m 𝑥)) ↔ 𝒫 𝑥 ≈ (2o ↑m 𝑥))) |
17 | 7, 16 | mpbiri 258 | . . 3 ⊢ (𝑥 ∈ Fin → (♯‘𝒫 𝑥) = (♯‘(2o ↑m 𝑥))) |
18 | hashmap 14395 | . . . . 5 ⊢ ((2o ∈ Fin ∧ 𝑥 ∈ Fin) → (♯‘(2o ↑m 𝑥)) = ((♯‘2o)↑(♯‘𝑥))) | |
19 | 12, 18 | mpan 689 | . . . 4 ⊢ (𝑥 ∈ Fin → (♯‘(2o ↑m 𝑥)) = ((♯‘2o)↑(♯‘𝑥))) |
20 | hash2 14365 | . . . . 5 ⊢ (♯‘2o) = 2 | |
21 | 20 | oveq1i 7419 | . . . 4 ⊢ ((♯‘2o)↑(♯‘𝑥)) = (2↑(♯‘𝑥)) |
22 | 19, 21 | eqtrdi 2789 | . . 3 ⊢ (𝑥 ∈ Fin → (♯‘(2o ↑m 𝑥)) = (2↑(♯‘𝑥))) |
23 | 17, 22 | eqtrd 2773 | . 2 ⊢ (𝑥 ∈ Fin → (♯‘𝒫 𝑥) = (2↑(♯‘𝑥))) |
24 | 5, 23 | vtoclga 3566 | 1 ⊢ (𝐴 ∈ Fin → (♯‘𝒫 𝐴) = (2↑(♯‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∅c0 4323 𝒫 cpw 4603 {csn 4629 {cpr 4631 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 2oc2o 8460 ↑m cmap 8820 ≈ cen 8936 Fincfn 8939 2c2 12267 ↑cexp 14027 ♯chash 14290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-seq 13967 df-exp 14028 df-hash 14291 |
This theorem is referenced by: ackbijnn 15774 |
Copyright terms: Public domain | W3C validator |