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Theorem sticksstones4 42151
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 2736 . . . . . 6 (𝑝𝐴 ↦ ran 𝑝) = (𝑝𝐴 ↦ ran 𝑝)
61, 2, 3, 4, 5sticksstones2 42149 . . . . 5 (𝜑 → (𝑝𝐴 ↦ ran 𝑝):𝐴1-1𝐵)
71, 2, 3, 4, 5sticksstones3 42150 . . . . 5 (𝜑 → (𝑝𝐴 ↦ ran 𝑝):𝐴onto𝐵)
86, 7jca 511 . . . 4 (𝜑 → ((𝑝𝐴 ↦ ran 𝑝):𝐴1-1𝐵 ∧ (𝑝𝐴 ↦ ran 𝑝):𝐴onto𝐵))
9 df-f1o 6567 . . . 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 4065 . . . . . . 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 14015 . . . . . . . 8 (𝜑 → (1...𝐾) ∈ Fin)
15 fzfid 14015 . . . . . . . 8 (𝜑 → (1...𝑁) ∈ Fin)
16 mapex 7964 . . . . . . . 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 5322 . . . . . . 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 2831 . . . . . 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 7245 . . . 4 (𝜑 → (𝑝𝐴 ↦ ran 𝑝) ∈ V)
23 f1oeq1 6835 . . . . . 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 3594 . . 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 8996 . 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 1539  wex 1778  wcel 2107  {cab 2713  wral 3060  {crab 3435  Vcvv 3479  wss 3950  𝒫 cpw 4599   class class class wbr 5142  cmpt 5224  ran crn 5685  wf 6556  1-1wf1 6557  ontowfo 6558  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  cen 8983  Fincfn 8986  1c1 11157   < clt 11296  0cn0 12528  ...cfz 13548  chash 14370
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-rep 5278  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  ax-pre-sup 11234
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-rmo 3379  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-int 4946  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-se 5637  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-isom 6569  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-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-oi 9551  df-card 9980  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  df-hash 14371
This theorem is referenced by:  sticksstones5  42152
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