Theorem hashinfom 11141

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 11139	. . . . 5 ⊢ ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
2	1	fveq1i 5671	. . . 4 ⊢ (♯‘𝐴) = (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))‘𝐴)
3		funmpt 5390	. . . . 5 ⊢ Fun (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
4		funrel 5369	. . . . . . 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 4716	. . . . . . 7 ⊢ ∅ ∈ ω
7		reldom 6980	. . . . . . . . . 10 ⊢ Rel ≼
8	7	brrelex2i 4794	. . . . . . . . 9 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
9		hashinfuni 11140	. . . . . . . . . 10 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω)
10		omex 4715	. . . . . . . . . 10 ⊢ ω ∈ V
11	9, 10	eqeltrdi 2323	. . . . . . . . 9 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ∈ V)
12		breq2 4113	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑦 ≼ 𝑥 ↔ 𝑦 ≼ 𝐴))
13	12	rabbidv 2802	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
14	13	unieqd 3925	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
15		eqid 2232	. . . . . . . . . 10 ⊢ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) = (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
16	14, 15	fvmptg 5753	. . . . . . . . 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 2265	. . . . . . 7 ⊢ (ω ≼ 𝐴 → ((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴) = ω)
19	6, 18	eleqtrrid 2322	. . . . . 6 ⊢ (ω ≼ 𝐴 → ∅ ∈ ((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴))
20		relelfvdm 5702	. . . . . 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 5747	. . . . 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 2277	. . 3 ⊢ (ω ≼ 𝐴 → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴)))
25	18	fveq2d 5674	. . 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 2265	. 2 ⊢ (ω ≼ 𝐴 → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω))
27		pnfxr 8326	. . 3 ⊢ +∞ ∈ ℝ*
28		ordom 4729	. . . . 5 ⊢ Ord ω
29		ordirr 4664	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
30	28, 29	ax-mp 5	. . . 4 ⊢ ¬ ω ∈ ω
31		zex 9586	. . . . . . . . . 10 ⊢ ℤ ∈ V
32	31	mptex 5912	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
33		vex 2816	. . . . . . . . 9 ⊢ 𝑧 ∈ V
34	32, 33	fvex 5690	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
35	34	ax-gen 1498	. . . . . . 7 ⊢ ∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
36		0z 9588	. . . . . . 7 ⊢ 0 ∈ ℤ
37		frecfnom 6632	. . . . . . 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 5455	. . . . . 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 2299	. . . 4 ⊢ (ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ↔ ω ∈ ω)
42	30, 41	mtbir 678	. . 3 ⊢ ¬ ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
43		fsnunfv 5885	. . 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 1374	. 2 ⊢ ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω) = +∞
45	26, 44	eqtrdi 2281	1 ⊢ (ω ≼ 𝐴 → (♯‘𝐴) = +∞)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1396   = wceq 1398   ∈ wcel 2203  {crab 2524  Vcvv 2813   ∪ cun 3209  ∅c0 3508  {csn 3689  ⟨cop 3692  ∪ cuni 3914   class class class wbr 4109   ↦ cmpt 4171  Ord word 4483  ωcom 4712  dom cdm 4749   ∘ ccom 4753  Rel wrel 4754  Fun wfun 5346   Fn wfn 5347  ‘cfv 5352  (class class class)co 6050  freccfrec 6621   ≼ cdom 6974  0cc0 8127  1c1 8128   + caddc 8130  +∞cpnf 8305  ℝ^*cxr 8307  ℤcz 9577  ♯chash 11138

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224  ax-rnegex 8236

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-recs 6536  df-frec 6622  df-dom 6977  df-pnf 8310  df-xr 8312  df-neg 8447  df-z 9578  df-ihash 11139

This theorem is referenced by:  filtinf  11154