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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > derang0 | Structured version Visualization version GIF version |
Description: The derangement number of the empty set. (Contributed by Mario Carneiro, 19-Jan-2015.) |
Ref | Expression |
---|---|
derang.d | ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) |
Ref | Expression |
---|---|
derang0 | ⊢ (𝐷‘∅) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0fin 9166 | . . 3 ⊢ ∅ ∈ Fin | |
2 | derang.d | . . . 4 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) | |
3 | 2 | derangval 34095 | . . 3 ⊢ (∅ ∈ Fin → (𝐷‘∅) = (♯‘{𝑓 ∣ (𝑓:∅–1-1-onto→∅ ∧ ∀𝑦 ∈ ∅ (𝑓‘𝑦) ≠ 𝑦)})) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (𝐷‘∅) = (♯‘{𝑓 ∣ (𝑓:∅–1-1-onto→∅ ∧ ∀𝑦 ∈ ∅ (𝑓‘𝑦) ≠ 𝑦)}) |
5 | ral0 4510 | . . . . . . 7 ⊢ ∀𝑦 ∈ ∅ (𝑓‘𝑦) ≠ 𝑦 | |
6 | 5 | biantru 531 | . . . . . 6 ⊢ (𝑓:∅–1-1-onto→∅ ↔ (𝑓:∅–1-1-onto→∅ ∧ ∀𝑦 ∈ ∅ (𝑓‘𝑦) ≠ 𝑦)) |
7 | eqid 2733 | . . . . . . 7 ⊢ ∅ = ∅ | |
8 | f1o00 6864 | . . . . . . 7 ⊢ (𝑓:∅–1-1-onto→∅ ↔ (𝑓 = ∅ ∧ ∅ = ∅)) | |
9 | 7, 8 | mpbiran2 709 | . . . . . 6 ⊢ (𝑓:∅–1-1-onto→∅ ↔ 𝑓 = ∅) |
10 | 6, 9 | bitr3i 277 | . . . . 5 ⊢ ((𝑓:∅–1-1-onto→∅ ∧ ∀𝑦 ∈ ∅ (𝑓‘𝑦) ≠ 𝑦) ↔ 𝑓 = ∅) |
11 | 10 | abbii 2803 | . . . 4 ⊢ {𝑓 ∣ (𝑓:∅–1-1-onto→∅ ∧ ∀𝑦 ∈ ∅ (𝑓‘𝑦) ≠ 𝑦)} = {𝑓 ∣ 𝑓 = ∅} |
12 | df-sn 4627 | . . . 4 ⊢ {∅} = {𝑓 ∣ 𝑓 = ∅} | |
13 | 11, 12 | eqtr4i 2764 | . . 3 ⊢ {𝑓 ∣ (𝑓:∅–1-1-onto→∅ ∧ ∀𝑦 ∈ ∅ (𝑓‘𝑦) ≠ 𝑦)} = {∅} |
14 | 13 | fveq2i 6890 | . 2 ⊢ (♯‘{𝑓 ∣ (𝑓:∅–1-1-onto→∅ ∧ ∀𝑦 ∈ ∅ (𝑓‘𝑦) ≠ 𝑦)}) = (♯‘{∅}) |
15 | 0ex 5305 | . . 3 ⊢ ∅ ∈ V | |
16 | hashsng 14324 | . . 3 ⊢ (∅ ∈ V → (♯‘{∅}) = 1) | |
17 | 15, 16 | ax-mp 5 | . 2 ⊢ (♯‘{∅}) = 1 |
18 | 4, 14, 17 | 3eqtri 2765 | 1 ⊢ (𝐷‘∅) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ∅c0 4320 {csn 4626 ↦ cmpt 5229 –1-1-onto→wf1o 6538 ‘cfv 6539 Fincfn 8934 1c1 11106 ♯chash 14285 |
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 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-n0 12468 df-z 12554 df-uz 12818 df-fz 13480 df-hash 14286 |
This theorem is referenced by: subfac0 34105 |
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