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Mirrors > Home > MPE Home > Th. List > birthdaylem1 | Structured version Visualization version GIF version |
Description: Lemma for birthday 25705. (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 6585 | . . . 4 ⊢ (𝑓:(1...𝐾)–1-1→(1...𝑁) → 𝑓:(1...𝐾)⟶(1...𝑁)) | |
2 | 1 | ss2abi 3966 | . . 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 3930 | . 2 ⊢ 𝑇 ⊆ 𝑆 |
6 | fzfi 13444 | . . . . 5 ⊢ (1...𝑁) ∈ Fin | |
7 | fzfi 13444 | . . . . 5 ⊢ (1...𝐾) ∈ Fin | |
8 | mapvalg 8460 | . . . . 5 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → ((1...𝑁) ↑m (1...𝐾)) = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}) | |
9 | 6, 7, 8 | mp2an 692 | . . . 4 ⊢ ((1...𝑁) ↑m (1...𝐾)) = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} |
10 | 4, 9 | eqtr4i 2765 | . . 3 ⊢ 𝑆 = ((1...𝑁) ↑m (1...𝐾)) |
11 | mapfi 8906 | . . . 4 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝐾) ∈ Fin) → ((1...𝑁) ↑m (1...𝐾)) ∈ Fin) | |
12 | 6, 7, 11 | mp2an 692 | . . 3 ⊢ ((1...𝑁) ↑m (1...𝐾)) ∈ Fin |
13 | 10, 12 | eqeltri 2830 | . 2 ⊢ 𝑆 ∈ Fin |
14 | elfz1end 13041 | . . . 4 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁)) | |
15 | ne0i 4233 | . . . 4 ⊢ (𝑁 ∈ (1...𝑁) → (1...𝑁) ≠ ∅) | |
16 | 14, 15 | sylbi 220 | . . 3 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ≠ ∅) |
17 | 10 | eqeq1i 2744 | . . . . 5 ⊢ (𝑆 = ∅ ↔ ((1...𝑁) ↑m (1...𝐾)) = ∅) |
18 | ovex 7216 | . . . . . . 7 ⊢ (1...𝑁) ∈ V | |
19 | ovex 7216 | . . . . . . 7 ⊢ (1...𝐾) ∈ V | |
20 | 18, 19 | map0 8510 | . . . . . 6 ⊢ (((1...𝑁) ↑m (1...𝐾)) = ∅ ↔ ((1...𝑁) = ∅ ∧ (1...𝐾) ≠ ∅)) |
21 | 20 | simplbi 501 | . . . . 5 ⊢ (((1...𝑁) ↑m (1...𝐾)) = ∅ → (1...𝑁) = ∅) |
22 | 17, 21 | sylbi 220 | . . . 4 ⊢ (𝑆 = ∅ → (1...𝑁) = ∅) |
23 | 22 | necon3i 2967 | . . 3 ⊢ ((1...𝑁) ≠ ∅ → 𝑆 ≠ ∅) |
24 | 16, 23 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑆 ≠ ∅) |
25 | 5, 13, 24 | 3pm3.2i 1340 | 1 ⊢ (𝑇 ⊆ 𝑆 ∧ 𝑆 ∈ Fin ∧ (𝑁 ∈ ℕ → 𝑆 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 {cab 2717 ≠ wne 2935 ⊆ wss 3853 ∅c0 4221 ⟶wf 6346 –1-1→wf1 6347 (class class class)co 7183 ↑m cmap 8450 Fincfn 8568 1c1 10629 ℕcn 11729 ...cfz 12994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-map 8452 df-pm 8453 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-n0 11990 df-z 12076 df-uz 12338 df-fz 12995 |
This theorem is referenced by: birthdaylem3 25704 birthday 25705 |
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