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Mirrors > Home > MPE Home > Th. List > hashfac | Structured version Visualization version GIF version |
Description: A factorial counts the number of bijections on a finite set. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by Mario Carneiro, 17-Apr-2015.) |
Ref | Expression |
---|---|
hashfac | ⊢ (𝐴 ∈ Fin → (♯‘{𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}) = (!‘(♯‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashf1 14492 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ Fin) → (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐴}) = ((!‘(♯‘𝐴)) · ((♯‘𝐴)C(♯‘𝐴)))) | |
2 | 1 | anidms 566 | . 2 ⊢ (𝐴 ∈ Fin → (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐴}) = ((!‘(♯‘𝐴)) · ((♯‘𝐴)C(♯‘𝐴)))) |
3 | enrefg 9022 | . . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ≈ 𝐴) | |
4 | f1finf1o 9302 | . . . . 5 ⊢ ((𝐴 ≈ 𝐴 ∧ 𝐴 ∈ Fin) → (𝑓:𝐴–1-1→𝐴 ↔ 𝑓:𝐴–1-1-onto→𝐴)) | |
5 | 3, 4 | mpancom 688 | . . . 4 ⊢ (𝐴 ∈ Fin → (𝑓:𝐴–1-1→𝐴 ↔ 𝑓:𝐴–1-1-onto→𝐴)) |
6 | 5 | abbidv 2805 | . . 3 ⊢ (𝐴 ∈ Fin → {𝑓 ∣ 𝑓:𝐴–1-1→𝐴} = {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}) |
7 | 6 | fveq2d 6910 | . 2 ⊢ (𝐴 ∈ Fin → (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐴}) = (♯‘{𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴})) |
8 | hashcl 14391 | . . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
9 | bcnn 14347 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴)C(♯‘𝐴)) = 1) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)C(♯‘𝐴)) = 1) |
11 | 10 | oveq2d 7446 | . . 3 ⊢ (𝐴 ∈ Fin → ((!‘(♯‘𝐴)) · ((♯‘𝐴)C(♯‘𝐴))) = ((!‘(♯‘𝐴)) · 1)) |
12 | 8 | faccld 14319 | . . . . 5 ⊢ (𝐴 ∈ Fin → (!‘(♯‘𝐴)) ∈ ℕ) |
13 | 12 | nncnd 12279 | . . . 4 ⊢ (𝐴 ∈ Fin → (!‘(♯‘𝐴)) ∈ ℂ) |
14 | 13 | mulridd 11275 | . . 3 ⊢ (𝐴 ∈ Fin → ((!‘(♯‘𝐴)) · 1) = (!‘(♯‘𝐴))) |
15 | 11, 14 | eqtrd 2774 | . 2 ⊢ (𝐴 ∈ Fin → ((!‘(♯‘𝐴)) · ((♯‘𝐴)C(♯‘𝐴))) = (!‘(♯‘𝐴))) |
16 | 2, 7, 15 | 3eqtr3d 2782 | 1 ⊢ (𝐴 ∈ Fin → (♯‘{𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}) = (!‘(♯‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 {cab 2711 class class class wbr 5147 –1-1→wf1 6559 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 ≈ cen 8980 Fincfn 8983 1c1 11153 · cmul 11157 ℕ0cn0 12523 !cfa 14308 Ccbc 14337 ♯chash 14365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-fz 13544 df-seq 14039 df-fac 14309 df-bc 14338 df-hash 14366 |
This theorem is referenced by: symghash 19409 subfaclefac 35160 poimirlem9 37615 |
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