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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 4555 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | fveq2d 6674 | . . 3 ⊢ (𝑥 = 𝐴 → (♯‘𝒫 𝑥) = (♯‘𝒫 𝐴)) |
3 | fveq2 6670 | . . . 4 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴)) | |
4 | 3 | oveq2d 7172 | . . 3 ⊢ (𝑥 = 𝐴 → (2↑(♯‘𝑥)) = (2↑(♯‘𝐴))) |
5 | 2, 4 | eqeq12d 2837 | . 2 ⊢ (𝑥 = 𝐴 → ((♯‘𝒫 𝑥) = (2↑(♯‘𝑥)) ↔ (♯‘𝒫 𝐴) = (2↑(♯‘𝐴)))) |
6 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | 6 | pw2en 8624 | . . . 4 ⊢ 𝒫 𝑥 ≈ (2o ↑m 𝑥) |
8 | pwfi 8819 | . . . . . 6 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin) | |
9 | 8 | biimpi 218 | . . . . 5 ⊢ (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin) |
10 | df2o2 8118 | . . . . . . 7 ⊢ 2o = {∅, {∅}} | |
11 | prfi 8793 | . . . . . . 7 ⊢ {∅, {∅}} ∈ Fin | |
12 | 10, 11 | eqeltri 2909 | . . . . . 6 ⊢ 2o ∈ Fin |
13 | mapfi 8820 | . . . . . 6 ⊢ ((2o ∈ Fin ∧ 𝑥 ∈ Fin) → (2o ↑m 𝑥) ∈ Fin) | |
14 | 12, 13 | mpan 688 | . . . . 5 ⊢ (𝑥 ∈ Fin → (2o ↑m 𝑥) ∈ Fin) |
15 | hashen 13708 | . . . . 5 ⊢ ((𝒫 𝑥 ∈ Fin ∧ (2o ↑m 𝑥) ∈ Fin) → ((♯‘𝒫 𝑥) = (♯‘(2o ↑m 𝑥)) ↔ 𝒫 𝑥 ≈ (2o ↑m 𝑥))) | |
16 | 9, 14, 15 | syl2anc 586 | . . . 4 ⊢ (𝑥 ∈ Fin → ((♯‘𝒫 𝑥) = (♯‘(2o ↑m 𝑥)) ↔ 𝒫 𝑥 ≈ (2o ↑m 𝑥))) |
17 | 7, 16 | mpbiri 260 | . . 3 ⊢ (𝑥 ∈ Fin → (♯‘𝒫 𝑥) = (♯‘(2o ↑m 𝑥))) |
18 | hashmap 13797 | . . . . 5 ⊢ ((2o ∈ Fin ∧ 𝑥 ∈ Fin) → (♯‘(2o ↑m 𝑥)) = ((♯‘2o)↑(♯‘𝑥))) | |
19 | 12, 18 | mpan 688 | . . . 4 ⊢ (𝑥 ∈ Fin → (♯‘(2o ↑m 𝑥)) = ((♯‘2o)↑(♯‘𝑥))) |
20 | hash2 13767 | . . . . 5 ⊢ (♯‘2o) = 2 | |
21 | 20 | oveq1i 7166 | . . . 4 ⊢ ((♯‘2o)↑(♯‘𝑥)) = (2↑(♯‘𝑥)) |
22 | 19, 21 | syl6eq 2872 | . . 3 ⊢ (𝑥 ∈ Fin → (♯‘(2o ↑m 𝑥)) = (2↑(♯‘𝑥))) |
23 | 17, 22 | eqtrd 2856 | . 2 ⊢ (𝑥 ∈ Fin → (♯‘𝒫 𝑥) = (2↑(♯‘𝑥))) |
24 | 5, 23 | vtoclga 3574 | 1 ⊢ (𝐴 ∈ Fin → (♯‘𝒫 𝐴) = (2↑(♯‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∅c0 4291 𝒫 cpw 4539 {csn 4567 {cpr 4569 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 2oc2o 8096 ↑m cmap 8406 ≈ cen 8506 Fincfn 8509 2c2 11693 ↑cexp 13430 ♯chash 13691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-seq 13371 df-exp 13431 df-hash 13692 |
This theorem is referenced by: ackbijnn 15183 |
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