Theorem sticksstones4 43113

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 2760	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 ↦ ran 𝑝) = (𝑝 ∈ 𝐴 ↦ ran 𝑝)
6	1, 2, 3, 4, 5	sticksstones2 43111	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1→𝐵)
7	1, 2, 3, 4, 5	sticksstones3 43112	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–onto→𝐵)
8	6, 7	jca 521	. . . 4 ⊢ (𝜑 → ((𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1→𝐵 ∧ (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–onto→𝐵))
9		df-f1o 6535	. . . 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 237	. . 3 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1-onto→𝐵)
11		simpl 488	. . . . . . . . 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 4013	. . . . . . 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 14070	. . . . . . . 8 ⊢ (𝜑 → (1...𝐾) ∈ Fin)
15		fzfid 14070	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
16		mapex 7936	. . . . . . . 8 ⊢ (((1...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V)
17	14, 15, 16	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V)
18		ssexg 5281	. . . . . . 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 596	. . . . . 6 ⊢ (𝜑 → {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ∈ V)
20	4	eleq1i 2851	. . . . . 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 237	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
22	21	mptexd 7219	. . . 4 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝) ∈ V)
23		f1oeq1 6801	. . . . . 6 ⊢ (𝑔 = (𝑝 ∈ 𝐴 ↦ ran 𝑝) → (𝑔:𝐴–1-1-onto→𝐵 ↔ (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1-onto→𝐵))
24	23	biimprd 251	. . . . 5 ⊢ (𝑔 = (𝑝 ∈ 𝐴 ↦ ran 𝑝) → ((𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1-onto→𝐵 → 𝑔:𝐴–1-1-onto→𝐵))
25	24	adantl 487	. . . 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 3549	. . 3 ⊢ (𝜑 → ((𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1-onto→𝐵 → ∃𝑔 𝑔:𝐴–1-1-onto→𝐵))
27	10, 26	mpd 16	. 2 ⊢ (𝜑 → ∃𝑔 𝑔:𝐴–1-1-onto→𝐵)
28		bren 8962	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵)
29	27, 28	sylibr 237	1 ⊢ (𝜑 → 𝐴 ≈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3899  𝒫 cpw 4557   class class class wbr 5103   ↦ cmpt 5186  ran crn 5649  ⟶wf 6524  –1-1→wf1 6525  –onto→wfo 6526  –1-1-onto→wf1o 6527  ‘cfv 6528  (class class class)co 7409   ≈ cen 8949  Fincfn 8952  1c1 11158   < clt 11300  ℕ₀cn0 12561  ...cfz 13594  ♯chash 14427

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735  ax-cnex 11213  ax-resscn 11214  ax-1cn 11215  ax-icn 11216  ax-addcl 11217  ax-addrcl 11218  ax-mulcl 11219  ax-mulrcl 11220  ax-mulcom 11221  ax-addass 11222  ax-mulass 11223  ax-distr 11224  ax-i2m1 11225  ax-1ne0 11226  ax-1rid 11227  ax-rnegex 11228  ax-rrecex 11229  ax-cnre 11230  ax-pre-lttri 11231  ax-pre-lttrn 11232  ax-pre-ltadd 11233  ax-pre-mulgt0 11234  ax-pre-sup 11235

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-isom 6537  df-riota 7366  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8455  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-inf 9413  df-oi 9482  df-card 9977  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-sub 11500  df-neg 11501  df-nn 12291  df-n0 12562  df-z 12649  df-uz 12921  df-fz 13595  df-hash 14428

This theorem is used by:  sticksstones5  43114