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| Description: Lemma for birthday 26998. (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 6803 | . . . 4 ⊢ (𝑓:(1...𝐾)–1-1→(1...𝑁) → 𝑓:(1...𝐾)⟶(1...𝑁)) | |
| 2 | 1 | ss2abi 4066 | . . 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 4034 | . 2 ⊢ 𝑇 ⊆ 𝑆 | 
| 6 | fzfi 14014 | . . . . 5 ⊢ (1...𝑁) ∈ Fin | |
| 7 | fzfi 14014 | . . . . 5 ⊢ (1...𝐾) ∈ Fin | |
| 8 | mapvalg 8877 | . . . . 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 2767 | . . 3 ⊢ 𝑆 = ((1...𝑁) ↑m (1...𝐾)) | 
| 11 | mapfi 9389 | . . . 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 2836 | . 2 ⊢ 𝑆 ∈ Fin | 
| 14 | elfz1end 13595 | . . . 4 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁)) | |
| 15 | ne0i 4340 | . . . 4 ⊢ (𝑁 ∈ (1...𝑁) → (1...𝑁) ≠ ∅) | |
| 16 | 14, 15 | sylbi 217 | . . 3 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ≠ ∅) | 
| 17 | 10 | eqeq1i 2741 | . . . . 5 ⊢ (𝑆 = ∅ ↔ ((1...𝑁) ↑m (1...𝐾)) = ∅) | 
| 18 | ovex 7465 | . . . . . . 7 ⊢ (1...𝑁) ∈ V | |
| 19 | ovex 7465 | . . . . . . 7 ⊢ (1...𝐾) ∈ V | |
| 20 | 18, 19 | map0 8928 | . . . . . 6 ⊢ (((1...𝑁) ↑m (1...𝐾)) = ∅ ↔ ((1...𝑁) = ∅ ∧ (1...𝐾) ≠ ∅)) | 
| 21 | 20 | simplbi 497 | . . . . 5 ⊢ (((1...𝑁) ↑m (1...𝐾)) = ∅ → (1...𝑁) = ∅) | 
| 22 | 17, 21 | sylbi 217 | . . . 4 ⊢ (𝑆 = ∅ → (1...𝑁) = ∅) | 
| 23 | 22 | necon3i 2972 | . . 3 ⊢ ((1...𝑁) ≠ ∅ → 𝑆 ≠ ∅) | 
| 24 | 16, 23 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑆 ≠ ∅) | 
| 25 | 5, 13, 24 | 3pm3.2i 1339 | 1 ⊢ (𝑇 ⊆ 𝑆 ∧ 𝑆 ∈ Fin ∧ (𝑁 ∈ ℕ → 𝑆 ≠ ∅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 {cab 2713 ≠ wne 2939 ⊆ wss 3950 ∅c0 4332 ⟶wf 6556 –1-1→wf1 6557 (class class class)co 7432 ↑m cmap 8867 Fincfn 8986 1c1 11157 ℕcn 12267 ...cfz 13548 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 | 
| This theorem is referenced by: birthdaylem3 26997 birthday 26998 | 
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