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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 4619 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | fveq2d 6911 | . . 3 ⊢ (𝑥 = 𝐴 → (♯‘𝒫 𝑥) = (♯‘𝒫 𝐴)) |
3 | fveq2 6907 | . . . 4 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴)) | |
4 | 3 | oveq2d 7447 | . . 3 ⊢ (𝑥 = 𝐴 → (2↑(♯‘𝑥)) = (2↑(♯‘𝐴))) |
5 | 2, 4 | eqeq12d 2751 | . 2 ⊢ (𝑥 = 𝐴 → ((♯‘𝒫 𝑥) = (2↑(♯‘𝑥)) ↔ (♯‘𝒫 𝐴) = (2↑(♯‘𝐴)))) |
6 | vex 3482 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | 6 | pw2en 9118 | . . . 4 ⊢ 𝒫 𝑥 ≈ (2o ↑m 𝑥) |
8 | pwfi 9355 | . . . . . 6 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin) | |
9 | 8 | biimpi 216 | . . . . 5 ⊢ (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin) |
10 | df2o2 8514 | . . . . . . 7 ⊢ 2o = {∅, {∅}} | |
11 | prfi 9361 | . . . . . . 7 ⊢ {∅, {∅}} ∈ Fin | |
12 | 10, 11 | eqeltri 2835 | . . . . . 6 ⊢ 2o ∈ Fin |
13 | mapfi 9386 | . . . . . 6 ⊢ ((2o ∈ Fin ∧ 𝑥 ∈ Fin) → (2o ↑m 𝑥) ∈ Fin) | |
14 | 12, 13 | mpan 690 | . . . . 5 ⊢ (𝑥 ∈ Fin → (2o ↑m 𝑥) ∈ Fin) |
15 | hashen 14383 | . . . . 5 ⊢ ((𝒫 𝑥 ∈ Fin ∧ (2o ↑m 𝑥) ∈ Fin) → ((♯‘𝒫 𝑥) = (♯‘(2o ↑m 𝑥)) ↔ 𝒫 𝑥 ≈ (2o ↑m 𝑥))) | |
16 | 9, 14, 15 | syl2anc 584 | . . . 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 14471 | . . . . 5 ⊢ ((2o ∈ Fin ∧ 𝑥 ∈ Fin) → (♯‘(2o ↑m 𝑥)) = ((♯‘2o)↑(♯‘𝑥))) | |
19 | 12, 18 | mpan 690 | . . . 4 ⊢ (𝑥 ∈ Fin → (♯‘(2o ↑m 𝑥)) = ((♯‘2o)↑(♯‘𝑥))) |
20 | hash2 14441 | . . . . 5 ⊢ (♯‘2o) = 2 | |
21 | 20 | oveq1i 7441 | . . . 4 ⊢ ((♯‘2o)↑(♯‘𝑥)) = (2↑(♯‘𝑥)) |
22 | 19, 21 | eqtrdi 2791 | . . 3 ⊢ (𝑥 ∈ Fin → (♯‘(2o ↑m 𝑥)) = (2↑(♯‘𝑥))) |
23 | 17, 22 | eqtrd 2775 | . 2 ⊢ (𝑥 ∈ Fin → (♯‘𝒫 𝑥) = (2↑(♯‘𝑥))) |
24 | 5, 23 | vtoclga 3577 | 1 ⊢ (𝐴 ∈ Fin → (♯‘𝒫 𝐴) = (2↑(♯‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∅c0 4339 𝒫 cpw 4605 {csn 4631 {cpr 4633 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 2oc2o 8499 ↑m cmap 8865 ≈ cen 8981 Fincfn 8984 2c2 12319 ↑cexp 14099 ♯chash 14366 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-seq 14040 df-exp 14100 df-hash 14367 |
This theorem is referenced by: ackbijnn 15861 |
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