Users' Mathboxes Mathbox for metakunt < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sticksstones4 Structured version   Visualization version   GIF version

Theorem sticksstones4 40603
Description: Equinumerosity lemma for sticks and stones. (Contributed by metakunt, 28-Sep-2024.)
Hypotheses
Ref Expression
sticksstones4.1 (𝜑𝑁 ∈ ℕ0)
sticksstones4.2 (𝜑𝐾 ∈ 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 → (𝑓𝑥) < (𝑓𝑦)))}
Assertion
Ref Expression
sticksstones4 (𝜑𝐴𝐵)
Distinct variable groups:   𝐴,𝑎   𝐴,𝑓   𝑥,𝐵,𝑦   𝐾,𝑎,𝑥,𝑦   𝑓,𝐾,𝑥,𝑦   𝑁,𝑎   𝑓,𝑁   𝜑,𝑎,𝑥,𝑦   𝜑,𝑓
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑓,𝑎)   𝑁(𝑥,𝑦)

Proof of Theorem sticksstones4
Dummy variables 𝑝 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sticksstones4.1 . . . . . 6 (𝜑𝑁 ∈ ℕ0)
2 sticksstones4.2 . . . . . 6 (𝜑𝐾 ∈ 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 → (𝑓𝑥) < (𝑓𝑦)))}
5 eqid 2733 . . . . . 6 (𝑝𝐴 ↦ ran 𝑝) = (𝑝𝐴 ↦ ran 𝑝)
61, 2, 3, 4, 5sticksstones2 40601 . . . . 5 (𝜑 → (𝑝𝐴 ↦ ran 𝑝):𝐴1-1𝐵)
71, 2, 3, 4, 5sticksstones3 40602 . . . . 5 (𝜑 → (𝑝𝐴 ↦ ran 𝑝):𝐴onto𝐵)
86, 7jca 513 . . . 4 (𝜑 → ((𝑝𝐴 ↦ ran 𝑝):𝐴1-1𝐵 ∧ (𝑝𝐴 ↦ ran 𝑝):𝐴onto𝐵))
9 df-f1o 6504 . . . 4 ((𝑝𝐴 ↦ ran 𝑝):𝐴1-1-onto𝐵 ↔ ((𝑝𝐴 ↦ ran 𝑝):𝐴1-1𝐵 ∧ (𝑝𝐴 ↦ ran 𝑝):𝐴onto𝐵))
108, 9sylibr 233 . . 3 (𝜑 → (𝑝𝐴 ↦ ran 𝑝):𝐴1-1-onto𝐵)
11 simpl 484 . . . . . . . . 9 ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))) → 𝑓:(1...𝐾)⟶(1...𝑁))
1211a1i 11 . . . . . . . 8 (𝜑 → ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))) → 𝑓:(1...𝐾)⟶(1...𝑁)))
1312ss2abdv 4021 . . . . . . 7 (𝜑 → {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))} ⊆ {𝑓𝑓:(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) → {𝑓𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V)
1714, 15, 16syl2anc 585 . . . . . . 7 (𝜑 → {𝑓𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V)
18 ssexg 5281 . . . . . . 7 (({𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))} ⊆ {𝑓𝑓:(1...𝐾)⟶(1...𝑁)} ∧ {𝑓𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V) → {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))} ∈ V)
1913, 17, 18syl2anc 585 . . . . . 6 (𝜑 → {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))} ∈ V)
204eleq1i 2825 . . . . . 6 (𝐴 ∈ V ↔ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))} ∈ V)
2119, 20sylibr 233 . . . . 5 (𝜑𝐴 ∈ V)
2221mptexd 7175 . . . 4 (𝜑 → (𝑝𝐴 ↦ ran 𝑝) ∈ V)
23 f1oeq1 6773 . . . . . 6 (𝑔 = (𝑝𝐴 ↦ ran 𝑝) → (𝑔:𝐴1-1-onto𝐵 ↔ (𝑝𝐴 ↦ ran 𝑝):𝐴1-1-onto𝐵))
2423biimprd 248 . . . . 5 (𝑔 = (𝑝𝐴 ↦ ran 𝑝) → ((𝑝𝐴 ↦ ran 𝑝):𝐴1-1-onto𝐵𝑔:𝐴1-1-onto𝐵))
2524adantl 483 . . . 4 ((𝜑𝑔 = (𝑝𝐴 ↦ ran 𝑝)) → ((𝑝𝐴 ↦ ran 𝑝):𝐴1-1-onto𝐵𝑔:𝐴1-1-onto𝐵))
2622, 25spcimedv 3553 . . 3 (𝜑 → ((𝑝𝐴 ↦ ran 𝑝):𝐴1-1-onto𝐵 → ∃𝑔 𝑔:𝐴1-1-onto𝐵))
2710, 26mpd 15 . 2 (𝜑 → ∃𝑔 𝑔:𝐴1-1-onto𝐵)
28 bren 8896 . 2 (𝐴𝐵 ↔ ∃𝑔 𝑔:𝐴1-1-onto𝐵)
2927, 28sylibr 233 1 (𝜑𝐴𝐵)
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  1-1wf1 6494  ontowfo 6495  1-1-ontowf1o 6496  cfv 6497  (class class class)co 7358  cen 8883  Fincfn 8886  1c1 11057   < clt 11194  0cn0 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
  Copyright terms: Public domain W3C validator