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Mirrors > Home > MPE Home > Th. List > Mathboxes > sticksstones4 | Structured version Visualization version GIF version |
Description: Equinumerosity lemma for sticks and stones. (Contributed by metakunt, 28-Sep-2024.) |
Ref | Expression |
---|---|
sticksstones4.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
sticksstones4.2 | ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
sticksstones4.3 | ⊢ 𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾} |
sticksstones4.4 | ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} |
Ref | Expression |
---|---|
sticksstones4 | ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sticksstones4.1 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
2 | sticksstones4.2 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
3 | sticksstones4.3 | . . . . . 6 ⊢ 𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾} | |
4 | sticksstones4.4 | . . . . . 6 ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} | |
5 | eqid 2733 | . . . . . 6 ⊢ (𝑝 ∈ 𝐴 ↦ ran 𝑝) = (𝑝 ∈ 𝐴 ↦ ran 𝑝) | |
6 | 1, 2, 3, 4, 5 | sticksstones2 40601 | . . . . 5 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1→𝐵) |
7 | 1, 2, 3, 4, 5 | sticksstones3 40602 | . . . . 5 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–onto→𝐵) |
8 | 6, 7 | jca 513 | . . . 4 ⊢ (𝜑 → ((𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1→𝐵 ∧ (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–onto→𝐵)) |
9 | df-f1o 6504 | . . . 4 ⊢ ((𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1-onto→𝐵 ↔ ((𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1→𝐵 ∧ (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–onto→𝐵)) | |
10 | 8, 9 | sylibr 233 | . . 3 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1-onto→𝐵) |
11 | simpl 484 | . . . . . . . . 9 ⊢ ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))) → 𝑓:(1...𝐾)⟶(1...𝑁)) | |
12 | 11 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))) → 𝑓:(1...𝐾)⟶(1...𝑁))) |
13 | 12 | ss2abdv 4021 | . . . . . . 7 ⊢ (𝜑 → {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ⊆ {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}) |
14 | fzfid 13884 | . . . . . . . 8 ⊢ (𝜑 → (1...𝐾) ∈ Fin) | |
15 | fzfid 13884 | . . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ∈ Fin) | |
16 | mapex 8774 | . . . . . . . 8 ⊢ (((1...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V) | |
17 | 14, 15, 16 | syl2anc 585 | . . . . . . 7 ⊢ (𝜑 → {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V) |
18 | ssexg 5281 | . . . . . . 7 ⊢ (({𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ⊆ {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} ∧ {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V) → {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ∈ V) | |
19 | 13, 17, 18 | syl2anc 585 | . . . . . 6 ⊢ (𝜑 → {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ∈ V) |
20 | 4 | eleq1i 2825 | . . . . . 6 ⊢ (𝐴 ∈ V ↔ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ∈ V) |
21 | 19, 20 | sylibr 233 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
22 | 21 | mptexd 7175 | . . . 4 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝) ∈ V) |
23 | f1oeq1 6773 | . . . . . 6 ⊢ (𝑔 = (𝑝 ∈ 𝐴 ↦ ran 𝑝) → (𝑔:𝐴–1-1-onto→𝐵 ↔ (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1-onto→𝐵)) | |
24 | 23 | biimprd 248 | . . . . 5 ⊢ (𝑔 = (𝑝 ∈ 𝐴 ↦ ran 𝑝) → ((𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1-onto→𝐵 → 𝑔:𝐴–1-1-onto→𝐵)) |
25 | 24 | adantl 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 = (𝑝 ∈ 𝐴 ↦ ran 𝑝)) → ((𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1-onto→𝐵 → 𝑔:𝐴–1-1-onto→𝐵)) |
26 | 22, 25 | spcimedv 3553 | . . 3 ⊢ (𝜑 → ((𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1-onto→𝐵 → ∃𝑔 𝑔:𝐴–1-1-onto→𝐵)) |
27 | 10, 26 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑔 𝑔:𝐴–1-1-onto→𝐵) |
28 | bren 8896 | . 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵) | |
29 | 27, 28 | sylibr 233 | 1 ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ∀wral 3061 {crab 3406 Vcvv 3444 ⊆ wss 3911 𝒫 cpw 4561 class class class wbr 5106 ↦ cmpt 5189 ran crn 5635 ⟶wf 6493 –1-1→wf1 6494 –onto→wfo 6495 –1-1-onto→wf1o 6496 ‘cfv 6497 (class class class)co 7358 ≈ cen 8883 Fincfn 8886 1c1 11057 < clt 11194 ℕ0cn0 12418 ...cfz 13430 ♯chash 14236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-hash 14237 |
This theorem is referenced by: sticksstones5 40604 |
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