Theorem hashinfom 10852

Description: The value of the ♯ function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.)

Assertion

Ref	Expression
hashinfom	⊢ (ω ≼ 𝐴 → (♯‘𝐴) = +∞)

Proof of Theorem hashinfom

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ihash 10850	. . . . 5 ⊢ ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
2	1	fveq1i 5556	. . . 4 ⊢ (♯‘𝐴) = (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))‘𝐴)
3		funmpt 5293	. . . . 5 ⊢ Fun (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
4		funrel 5272	. . . . . . 7 ⊢ (Fun (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) → Rel (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
5	3, 4	ax-mp 5	. . . . . 6 ⊢ Rel (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
6		peano1 4627	. . . . . . 7 ⊢ ∅ ∈ ω
7		reldom 6801	. . . . . . . . . 10 ⊢ Rel ≼
8	7	brrelex2i 4704	. . . . . . . . 9 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
9		hashinfuni 10851	. . . . . . . . . 10 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω)
10		omex 4626	. . . . . . . . . 10 ⊢ ω ∈ V
11	9, 10	eqeltrdi 2284	. . . . . . . . 9 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ∈ V)
12		breq2 4034	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑦 ≼ 𝑥 ↔ 𝑦 ≼ 𝐴))
13	12	rabbidv 2749	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
14	13	unieqd 3847	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
15		eqid 2193	. . . . . . . . . 10 ⊢ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) = (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
16	14, 15	fvmptg 5634	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ∈ V) → ((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴) = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
17	8, 11, 16	syl2anc 411	. . . . . . . 8 ⊢ (ω ≼ 𝐴 → ((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴) = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
18	17, 9	eqtrd 2226	. . . . . . 7 ⊢ (ω ≼ 𝐴 → ((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴) = ω)
19	6, 18	eleqtrrid 2283	. . . . . 6 ⊢ (ω ≼ 𝐴 → ∅ ∈ ((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴))
20		relelfvdm 5587	. . . . . 6 ⊢ ((Rel (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) ∧ ∅ ∈ ((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴)) → 𝐴 ∈ dom (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
21	5, 19, 20	sylancr 414	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ dom (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
22		fvco 5628	. . . . 5 ⊢ ((Fun (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) ∧ 𝐴 ∈ dom (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})) → (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴)))
23	3, 21, 22	sylancr 414	. . . 4 ⊢ (ω ≼ 𝐴 → (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴)))
24	2, 23	eqtrid 2238	. . 3 ⊢ (ω ≼ 𝐴 → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴)))
25	18	fveq2d 5559	. . 3 ⊢ (ω ≼ 𝐴 → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴)) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω))
26	24, 25	eqtrd 2226	. 2 ⊢ (ω ≼ 𝐴 → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω))
27		pnfxr 8074	. . 3 ⊢ +∞ ∈ ℝ*
28		ordom 4640	. . . . 5 ⊢ Ord ω
29		ordirr 4575	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
30	28, 29	ax-mp 5	. . . 4 ⊢ ¬ ω ∈ ω
31		zex 9329	. . . . . . . . . 10 ⊢ ℤ ∈ V
32	31	mptex 5785	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
33		vex 2763	. . . . . . . . 9 ⊢ 𝑧 ∈ V
34	32, 33	fvex 5575	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
35	34	ax-gen 1460	. . . . . . 7 ⊢ ∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
36		0z 9331	. . . . . . 7 ⊢ 0 ∈ ℤ
37		frecfnom 6456	. . . . . . 7 ⊢ ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 0 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
38	35, 36, 37	mp2an 426	. . . . . 6 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω
39		fndm 5354	. . . . . 6 ⊢ (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω → dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) = ω)
40	38, 39	ax-mp 5	. . . . 5 ⊢ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) = ω
41	40	eleq2i 2260	. . . 4 ⊢ (ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ↔ ω ∈ ω)
42	30, 41	mtbir 672	. . 3 ⊢ ¬ ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
43		fsnunfv 5760	. . 3 ⊢ ((ω ∈ V ∧ +∞ ∈ ℝ* ∧ ¬ ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)) → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω) = +∞)
44	10, 27, 42, 43	mp3an 1348	. 2 ⊢ ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω) = +∞
45	26, 44	eqtrdi 2242	1 ⊢ (ω ≼ 𝐴 → (♯‘𝐴) = +∞)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1362   = wceq 1364   ∈ wcel 2164  {crab 2476  Vcvv 2760   ∪ cun 3152  ∅c0 3447  {csn 3619  ⟨cop 3622  ∪ cuni 3836   class class class wbr 4030   ↦ cmpt 4091  Ord word 4394  ωcom 4623  dom cdm 4660   ∘ ccom 4664  Rel wrel 4665  Fun wfun 5249   Fn wfn 5250  ‘cfv 5255  (class class class)co 5919  freccfrec 6445   ≼ cdom 6795  0cc0 7874  1c1 7875   + caddc 7877  +∞cpnf 8053  ℝ^*cxr 8055  ℤcz 9320  ♯chash 10849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971  ax-rnegex 7983

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-recs 6360  df-frec 6446  df-dom 6798  df-pnf 8058  df-xr 8060  df-neg 8195  df-z 9321  df-ihash 10850

This theorem is referenced by:  filtinf  10865