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Mirrors > Home > MPE Home > Th. List > birthdaylem1 | Structured version Visualization version GIF version |
Description: Lemma for birthday 25526. (Contributed by Mario Carneiro, 17-Apr-2015.) |
Ref | Expression |
---|---|
birthday.s | ⊢ 𝑆 = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} |
birthday.t | ⊢ 𝑇 = {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)} |
Ref | Expression |
---|---|
birthdaylem1 | ⊢ (𝑇 ⊆ 𝑆 ∧ 𝑆 ∈ Fin ∧ (𝑁 ∈ ℕ → 𝑆 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6569 | . . . 4 ⊢ (𝑓:(1...𝐾)–1-1→(1...𝑁) → 𝑓:(1...𝐾)⟶(1...𝑁)) | |
2 | 1 | ss2abi 4042 | . . 3 ⊢ {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} |
3 | birthday.t | . . 3 ⊢ 𝑇 = {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)} | |
4 | birthday.s | . . 3 ⊢ 𝑆 = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} | |
5 | 2, 3, 4 | 3sstr4i 4009 | . 2 ⊢ 𝑇 ⊆ 𝑆 |
6 | fzfi 13334 | . . . . 5 ⊢ (1...𝑁) ∈ Fin | |
7 | fzfi 13334 | . . . . 5 ⊢ (1...𝐾) ∈ Fin | |
8 | mapvalg 8410 | . . . . 5 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → ((1...𝑁) ↑m (1...𝐾)) = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}) | |
9 | 6, 7, 8 | mp2an 690 | . . . 4 ⊢ ((1...𝑁) ↑m (1...𝐾)) = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} |
10 | 4, 9 | eqtr4i 2847 | . . 3 ⊢ 𝑆 = ((1...𝑁) ↑m (1...𝐾)) |
11 | mapfi 8814 | . . . 4 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → ((1...𝑁) ↑m (1...𝐾)) ∈ Fin) | |
12 | 6, 7, 11 | mp2an 690 | . . 3 ⊢ ((1...𝑁) ↑m (1...𝐾)) ∈ Fin |
13 | 10, 12 | eqeltri 2909 | . 2 ⊢ 𝑆 ∈ Fin |
14 | elfz1end 12931 | . . . 4 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁)) | |
15 | ne0i 4299 | . . . 4 ⊢ (𝑁 ∈ (1...𝑁) → (1...𝑁) ≠ ∅) | |
16 | 14, 15 | sylbi 219 | . . 3 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ≠ ∅) |
17 | 10 | eqeq1i 2826 | . . . . 5 ⊢ (𝑆 = ∅ ↔ ((1...𝑁) ↑m (1...𝐾)) = ∅) |
18 | ovex 7183 | . . . . . . 7 ⊢ (1...𝑁) ∈ V | |
19 | ovex 7183 | . . . . . . 7 ⊢ (1...𝐾) ∈ V | |
20 | 18, 19 | map0 8445 | . . . . . 6 ⊢ (((1...𝑁) ↑m (1...𝐾)) = ∅ ↔ ((1...𝑁) = ∅ ∧ (1...𝐾) ≠ ∅)) |
21 | 20 | simplbi 500 | . . . . 5 ⊢ (((1...𝑁) ↑m (1...𝐾)) = ∅ → (1...𝑁) = ∅) |
22 | 17, 21 | sylbi 219 | . . . 4 ⊢ (𝑆 = ∅ → (1...𝑁) = ∅) |
23 | 22 | necon3i 3048 | . . 3 ⊢ ((1...𝑁) ≠ ∅ → 𝑆 ≠ ∅) |
24 | 16, 23 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑆 ≠ ∅) |
25 | 5, 13, 24 | 3pm3.2i 1335 | 1 ⊢ (𝑇 ⊆ 𝑆 ∧ 𝑆 ∈ Fin ∧ (𝑁 ∈ ℕ → 𝑆 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ⊆ wss 3935 ∅c0 4290 ⟶wf 6345 –1-1→wf1 6346 (class class class)co 7150 ↑m cmap 8400 Fincfn 8503 1c1 10532 ℕcn 11632 ...cfz 12886 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 |
This theorem is referenced by: birthdaylem3 25525 birthday 25526 |
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