Theorem sticksstones4 41424

Description: Equinumerosity lemma for sticks and stones. (Contributed by metakunt, 28-Sep-2024.)

Hypotheses

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)))}

Assertion

Ref	Expression
sticksstones4	⊢ (𝜑 → 𝐴 ≈ 𝐵)

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.

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 2724	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 ↦ ran 𝑝) = (𝑝 ∈ 𝐴 ↦ ran 𝑝)
6	1, 2, 3, 4, 5	sticksstones2 41422	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1→𝐵)
7	1, 2, 3, 4, 5	sticksstones3 41423	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–onto→𝐵)
8	6, 7	jca 511	. . . 4 ⊢ (𝜑 → ((𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1→𝐵 ∧ (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–onto→𝐵))
9		df-f1o 6540	. . . 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 482	. . . . . . . . 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 4052	. . . . . . 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 13934	. . . . . . . 8 ⊢ (𝜑 → (1...𝐾) ∈ Fin)
15		fzfid 13934	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
16		mapex 8821	. . . . . . . 8 ⊢ (((1...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V)
17	14, 15, 16	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V)
18		ssexg 5313	. . . . . . 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 583	. . . . . 6 ⊢ (𝜑 → {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ∈ V)
20	4	eleq1i 2816	. . . . . 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 7217	. . . 4 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝) ∈ V)
23		f1oeq1 6811	. . . . . 6 ⊢ (𝑔 = (𝑝 ∈ 𝐴 ↦ ran 𝑝) → (𝑔:𝐴–1-1-onto→𝐵 ↔ (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1-onto→𝐵))
24	23	biimprd 247	. . . . 5 ⊢ (𝑔 = (𝑝 ∈ 𝐴 ↦ ran 𝑝) → ((𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1-onto→𝐵 → 𝑔:𝐴–1-1-onto→𝐵))
25	24	adantl 481	. . . 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 3577	. . 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 8944	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵)
29	27, 28	sylibr 233	1 ⊢ (𝜑 → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2701  ∀wral 3053  {crab 3424  Vcvv 3466   ⊆ wss 3940  𝒫 cpw 4594   class class class wbr 5138   ↦ cmpt 5221  ran crn 5667  ⟶wf 6529  –1-1→wf1 6530  –onto→wfo 6531  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7401   ≈ cen 8931  Fincfn 8934  1c1 11106   < clt 11244  ℕ₀cn0 12468  ...cfz 13480  ♯chash 14286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-sup 9432  df-inf 9433  df-oi 9500  df-card 9929  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-hash 14287

This theorem is referenced by:  sticksstones5  41425