Theorem sticksstones4 41673

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 2725	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 ↦ ran 𝑝) = (𝑝 ∈ 𝐴 ↦ ran 𝑝)
6	1, 2, 3, 4, 5	sticksstones2 41671	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1→𝐵)
7	1, 2, 3, 4, 5	sticksstones3 41672	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–onto→𝐵)
8	6, 7	jca 510	. . . 4 ⊢ (𝜑 → ((𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–1-1→𝐵 ∧ (𝑝 ∈ 𝐴 ↦ ran 𝑝):𝐴–onto→𝐵))
9		df-f1o 6550	. . . 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 481	. . . . . . . . 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 4054	. . . . . . 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 13965	. . . . . . . 8 ⊢ (𝜑 → (1...𝐾) ∈ Fin)
15		fzfid 13965	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
16		mapex 8844	. . . . . . . 8 ⊢ (((1...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V)
17	14, 15, 16	syl2anc 582	. . . . . . 7 ⊢ (𝜑 → {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)} ∈ V)
18		ssexg 5319	. . . . . . 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 582	. . . . . 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 7230	. . . 4 ⊢ (𝜑 → (𝑝 ∈ 𝐴 ↦ ran 𝑝) ∈ V)
23		f1oeq1 6820	. . . . . 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 480	. . . 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 3576	. . 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 8967	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵)
29	27, 28	sylibr 233	1 ⊢ (𝜑 → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702  ∀wral 3051  {crab 3419  Vcvv 3463   ⊆ wss 3941  𝒫 cpw 4599   class class class wbr 5144   ↦ cmpt 5227  ran crn 5674  ⟶wf 6539  –1-1→wf1 6540  –onto→wfo 6541  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7413   ≈ cen 8954  Fincfn 8957  1c1 11134   < clt 11273  ℕ₀cn0 12497  ...cfz 13511  ♯chash 14316

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 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-pre-sup 11211

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  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 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9460  df-inf 9461  df-oi 9528  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-n0 12498  df-z 12584  df-uz 12848  df-fz 13512  df-hash 14317

This theorem is referenced by:  sticksstones5  41674