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Theorem sticksstones4 41553
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 2727 . . . . . 6 (𝑝𝐴 ↦ ran 𝑝) = (𝑝𝐴 ↦ ran 𝑝)
61, 2, 3, 4, 5sticksstones2 41551 . . . . 5 (𝜑 → (𝑝𝐴 ↦ ran 𝑝):𝐴1-1𝐵)
71, 2, 3, 4, 5sticksstones3 41552 . . . . 5 (𝜑 → (𝑝𝐴 ↦ ran 𝑝):𝐴onto𝐵)
86, 7jca 511 . . . 4 (𝜑 → ((𝑝𝐴 ↦ ran 𝑝):𝐴1-1𝐵 ∧ (𝑝𝐴 ↦ ran 𝑝):𝐴onto𝐵))
9 df-f1o 6549 . . . 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 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 4056 . . . . . . 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 13962 . . . . . . . 8 (𝜑 → (1...𝐾) ∈ Fin)
15 fzfid 13962 . . . . . . . 8 (𝜑 → (1...𝑁) ∈ Fin)
16 mapex 8842 . . . . . . . 8 (((1...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → {𝑓𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V)
1714, 15, 16syl2anc 583 . . . . . . 7 (𝜑 → {𝑓𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V)
18 ssexg 5317 . . . . . . 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 583 . . . . . 6 (𝜑 → {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))} ∈ V)
204eleq1i 2819 . . . . . 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 7230 . . . 4 (𝜑 → (𝑝𝐴 ↦ ran 𝑝) ∈ V)
23 f1oeq1 6821 . . . . . 6 (𝑔 = (𝑝𝐴 ↦ ran 𝑝) → (𝑔:𝐴1-1-onto𝐵 ↔ (𝑝𝐴 ↦ ran 𝑝):𝐴1-1-onto𝐵))
2423biimprd 247 . . . . 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 3580 . . 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 8965 . 2 (𝐴𝐵 ↔ ∃𝑔 𝑔:𝐴1-1-onto𝐵)
2927, 28sylibr 233 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wex 1774  wcel 2099  {cab 2704  wral 3056  {crab 3427  Vcvv 3469  wss 3944  𝒫 cpw 4598   class class class wbr 5142  cmpt 5225  ran crn 5673  wf 6538  1-1wf1 6539  ontowfo 6540  1-1-ontowf1o 6541  cfv 6542  (class class class)co 7414  cen 8952  Fincfn 8955  1c1 11131   < clt 11270  0cn0 12494  ...cfz 13508  chash 14313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-oi 9525  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-n0 12495  df-z 12581  df-uz 12845  df-fz 13509  df-hash 14314
This theorem is referenced by:  sticksstones5  41554
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