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Mirrors > Home > MPE Home > Th. List > birthdaylem1 | Structured version Visualization version GIF version |
Description: Lemma for birthday 25534. (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 6577 | . . . 4 ⊢ (𝑓:(1...𝐾)–1-1→(1...𝑁) → 𝑓:(1...𝐾)⟶(1...𝑁)) | |
2 | 1 | ss2abi 4045 | . . 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 4012 | . 2 ⊢ 𝑇 ⊆ 𝑆 |
6 | fzfi 13343 | . . . . 5 ⊢ (1...𝑁) ∈ Fin | |
7 | fzfi 13343 | . . . . 5 ⊢ (1...𝐾) ∈ Fin | |
8 | mapvalg 8418 | . . . . 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 2849 | . . 3 ⊢ 𝑆 = ((1...𝑁) ↑m (1...𝐾)) |
11 | mapfi 8822 | . . . 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 2911 | . 2 ⊢ 𝑆 ∈ Fin |
14 | elfz1end 12940 | . . . 4 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁)) | |
15 | ne0i 4302 | . . . 4 ⊢ (𝑁 ∈ (1...𝑁) → (1...𝑁) ≠ ∅) | |
16 | 14, 15 | sylbi 219 | . . 3 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ≠ ∅) |
17 | 10 | eqeq1i 2828 | . . . . 5 ⊢ (𝑆 = ∅ ↔ ((1...𝑁) ↑m (1...𝐾)) = ∅) |
18 | ovex 7191 | . . . . . . 7 ⊢ (1...𝑁) ∈ V | |
19 | ovex 7191 | . . . . . . 7 ⊢ (1...𝐾) ∈ V | |
20 | 18, 19 | map0 8453 | . . . . . 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 3050 | . . 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 1537 ∈ wcel 2114 {cab 2801 ≠ wne 3018 ⊆ wss 3938 ∅c0 4293 ⟶wf 6353 –1-1→wf1 6354 (class class class)co 7158 ↑m cmap 8408 Fincfn 8511 1c1 10540 ℕcn 11640 ...cfz 12895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 |
This theorem is referenced by: birthdaylem3 25533 birthday 25534 |
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