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Mirrors > Home > MPE Home > Th. List > birthdaylem1 | Structured version Visualization version GIF version |
Description: Lemma for birthday 24726. (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 6139 | . . . 4 ⊢ (𝑓:(1...𝐾)–1-1→(1...𝑁) → 𝑓:(1...𝐾)⟶(1...𝑁)) | |
2 | 1 | ss2abi 3707 | . . 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 3677 | . 2 ⊢ 𝑇 ⊆ 𝑆 |
6 | fzfi 12811 | . . . . 5 ⊢ (1...𝑁) ∈ Fin | |
7 | fzfi 12811 | . . . . 5 ⊢ (1...𝐾) ∈ Fin | |
8 | mapvalg 7909 | . . . . 5 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → ((1...𝑁) ↑𝑚 (1...𝐾)) = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}) | |
9 | 6, 7, 8 | mp2an 708 | . . . 4 ⊢ ((1...𝑁) ↑𝑚 (1...𝐾)) = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} |
10 | 4, 9 | eqtr4i 2676 | . . 3 ⊢ 𝑆 = ((1...𝑁) ↑𝑚 (1...𝐾)) |
11 | mapfi 8303 | . . . 4 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → ((1...𝑁) ↑𝑚 (1...𝐾)) ∈ Fin) | |
12 | 6, 7, 11 | mp2an 708 | . . 3 ⊢ ((1...𝑁) ↑𝑚 (1...𝐾)) ∈ Fin |
13 | 10, 12 | eqeltri 2726 | . 2 ⊢ 𝑆 ∈ Fin |
14 | elfz1end 12409 | . . . 4 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁)) | |
15 | ne0i 3954 | . . . 4 ⊢ (𝑁 ∈ (1...𝑁) → (1...𝑁) ≠ ∅) | |
16 | 14, 15 | sylbi 207 | . . 3 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ≠ ∅) |
17 | 10 | eqeq1i 2656 | . . . . 5 ⊢ (𝑆 = ∅ ↔ ((1...𝑁) ↑𝑚 (1...𝐾)) = ∅) |
18 | ovex 6718 | . . . . . . 7 ⊢ (1...𝑁) ∈ V | |
19 | ovex 6718 | . . . . . . 7 ⊢ (1...𝐾) ∈ V | |
20 | 18, 19 | map0 7940 | . . . . . 6 ⊢ (((1...𝑁) ↑𝑚 (1...𝐾)) = ∅ ↔ ((1...𝑁) = ∅ ∧ (1...𝐾) ≠ ∅)) |
21 | 20 | simplbi 475 | . . . . 5 ⊢ (((1...𝑁) ↑𝑚 (1...𝐾)) = ∅ → (1...𝑁) = ∅) |
22 | 17, 21 | sylbi 207 | . . . 4 ⊢ (𝑆 = ∅ → (1...𝑁) = ∅) |
23 | 22 | necon3i 2855 | . . 3 ⊢ ((1...𝑁) ≠ ∅ → 𝑆 ≠ ∅) |
24 | 16, 23 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑆 ≠ ∅) |
25 | 5, 13, 24 | 3pm3.2i 1259 | 1 ⊢ (𝑇 ⊆ 𝑆 ∧ 𝑆 ∈ Fin ∧ (𝑁 ∈ ℕ → 𝑆 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 {cab 2637 ≠ wne 2823 ⊆ wss 3607 ∅c0 3948 ⟶wf 5922 –1-1→wf1 5923 (class class class)co 6690 ↑𝑚 cmap 7899 Fincfn 7997 1c1 9975 ℕcn 11058 ...cfz 12364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 |
This theorem is referenced by: birthdaylem3 24725 birthday 24726 |
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