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Theorem sticksstones4 42131
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 2735 . . . . . 6 (𝑝𝐴 ↦ ran 𝑝) = (𝑝𝐴 ↦ ran 𝑝)
61, 2, 3, 4, 5sticksstones2 42129 . . . . 5 (𝜑 → (𝑝𝐴 ↦ ran 𝑝):𝐴1-1𝐵)
71, 2, 3, 4, 5sticksstones3 42130 . . . . 5 (𝜑 → (𝑝𝐴 ↦ ran 𝑝):𝐴onto𝐵)
86, 7jca 511 . . . 4 (𝜑 → ((𝑝𝐴 ↦ ran 𝑝):𝐴1-1𝐵 ∧ (𝑝𝐴 ↦ ran 𝑝):𝐴onto𝐵))
9 df-f1o 6570 . . . 4 ((𝑝𝐴 ↦ ran 𝑝):𝐴1-1-onto𝐵 ↔ ((𝑝𝐴 ↦ ran 𝑝):𝐴1-1𝐵 ∧ (𝑝𝐴 ↦ ran 𝑝):𝐴onto𝐵))
108, 9sylibr 234 . . 3 (𝜑 → (𝑝𝐴 ↦ ran 𝑝):𝐴1-1-onto𝐵)
11 simpl 482 . . . . . . . . 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 4076 . . . . . . 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 14011 . . . . . . . 8 (𝜑 → (1...𝐾) ∈ Fin)
15 fzfid 14011 . . . . . . . 8 (𝜑 → (1...𝑁) ∈ Fin)
16 mapex 7962 . . . . . . . 8 (((1...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → {𝑓𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V)
1714, 15, 16syl2anc 584 . . . . . . 7 (𝜑 → {𝑓𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V)
18 ssexg 5329 . . . . . . 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 584 . . . . . 6 (𝜑 → {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))} ∈ V)
204eleq1i 2830 . . . . . 6 (𝐴 ∈ V ↔ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))} ∈ V)
2119, 20sylibr 234 . . . . 5 (𝜑𝐴 ∈ V)
2221mptexd 7244 . . . 4 (𝜑 → (𝑝𝐴 ↦ ran 𝑝) ∈ V)
23 f1oeq1 6837 . . . . . 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 481 . . . 4 ((𝜑𝑔 = (𝑝𝐴 ↦ ran 𝑝)) → ((𝑝𝐴 ↦ ran 𝑝):𝐴1-1-onto𝐵𝑔:𝐴1-1-onto𝐵))
2622, 25spcimedv 3595 . . 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 8994 . 2 (𝐴𝐵 ↔ ∃𝑔 𝑔:𝐴1-1-onto𝐵)
2927, 28sylibr 234 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605   class class class wbr 5148  cmpt 5231  ran crn 5690  wf 6559  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  cen 8981  Fincfn 8984  1c1 11154   < clt 11293  0cn0 12524  ...cfz 13544  chash 14366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367
This theorem is referenced by:  sticksstones5  42132
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