Theorem hashennn 10989

Description: The size of a set equinumerous to an element of ω. (Contributed by Jim Kingdon, 21-Feb-2022.)

Assertion

Ref	Expression
hashennn	⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (♯‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑁

Proof of Theorem hashennn

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

Step	Hyp	Ref	Expression
1		df-ihash 10985	. . . . 5 ⊢ ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
2	1	fveq1i 5624	. . . 4 ⊢ (♯‘𝐴) = (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))‘𝐴)
3		funmpt 5352	. . . . 5 ⊢ Fun (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
4		hashennnuni 10988	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = 𝑁)
5	4	eqcomd 2235	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝑁 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
6		nnfi 7022	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → 𝑁 ∈ Fin)
7	6	adantr 276	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝑁 ∈ Fin)
8		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝑁 ≈ 𝐴)
9	8	ensymd 6925	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝐴 ≈ 𝑁)
10		enfii 7024	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝐴 ≈ 𝑁) → 𝐴 ∈ Fin)
11	7, 9, 10	syl2anc 411	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝐴 ∈ Fin)
12		simpl 109	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝑁 ∈ ω)
13		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → 𝑧 = 𝑁)
14		breq2 4086	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → (𝑦 ≼ 𝑥 ↔ 𝑦 ≼ 𝐴))
15	14	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → (𝑦 ≼ 𝑥 ↔ 𝑦 ≼ 𝐴))
16	15	rabbidv 2788	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
17	16	unieqd 3898	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
18	13, 17	eqeq12d 2244	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → (𝑧 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} ↔ 𝑁 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}))
19	18	opelopabga 4350	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝑁 ∈ ω) → (⟨𝐴, 𝑁⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}} ↔ 𝑁 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}))
20	11, 12, 19	syl2anc 411	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (⟨𝐴, 𝑁⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}} ↔ 𝑁 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}))
21	5, 20	mpbird 167	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ⟨𝐴, 𝑁⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}})
22		mptv 4180	. . . . . . 7 ⊢ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}}
23	21, 22	eleqtrrdi 2323	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ⟨𝐴, 𝑁⟩ ∈ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
24		opeldmg 4925	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝑁 ∈ ω) → (⟨𝐴, 𝑁⟩ ∈ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) → 𝐴 ∈ dom (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})))
25	11, 12, 24	syl2anc 411	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (⟨𝐴, 𝑁⟩ ∈ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) → 𝐴 ∈ dom (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})))
26	23, 25	mpd 13	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝐴 ∈ dom (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
27		fvco 5697	. . . . 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)))
28	3, 26, 27	sylancr 414	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 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)))
29	2, 28	eqtrid 2274	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴)))
30	11	elexd 2813	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝐴 ∈ V)
31	4, 12	eqeltrd 2306	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ∈ ω)
32	14	rabbidv 2788	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
33	32	unieqd 3898	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
34		eqid 2229	. . . . . . 7 ⊢ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) = (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
35	33, 34	fvmptg 5703	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ∈ ω) → ((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴) = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
36	30, 31, 35	syl2anc 411	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴) = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
37	36, 4	eqtrd 2262	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴) = 𝑁)
38	37	fveq2d 5627	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴)) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘𝑁))
39	29, 38	eqtrd 2262	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘𝑁))
40		ordom 4696	. . . . . . 7 ⊢ Ord ω
41		ordirr 4631	. . . . . . 7 ⊢ (Ord ω → ¬ ω ∈ ω)
42	40, 41	ax-mp 5	. . . . . 6 ⊢ ¬ ω ∈ ω
43		eleq1 2292	. . . . . 6 ⊢ (ω = 𝑁 → (ω ∈ ω ↔ 𝑁 ∈ ω))
44	42, 43	mtbii 678	. . . . 5 ⊢ (ω = 𝑁 → ¬ 𝑁 ∈ ω)
45	44	necon2ai 2454	. . . 4 ⊢ (𝑁 ∈ ω → ω ≠ 𝑁)
46		fvunsng 5826	. . . 4 ⊢ ((𝑁 ∈ ω ∧ ω ≠ 𝑁) → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘𝑁) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))
47	45, 46	mpdan 421	. . 3 ⊢ (𝑁 ∈ ω → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘𝑁) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))
48	47	adantr 276	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘𝑁) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))
49	39, 48	eqtrd 2262	1 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (♯‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  {crab 2512  Vcvv 2799   ∪ cun 3195  {csn 3666  ⟨cop 3669  ∪ cuni 3887   class class class wbr 4082  {copab 4143   ↦ cmpt 4144  Ord word 4450  ωcom 4679  dom cdm 4716   ∘ ccom 4720  Fun wfun 5308  ‘cfv 5314  (class class class)co 5994  freccfrec 6526   ≈ cen 6875   ≼ cdom 6876  Fincfn 6877  0cc0 7987  1c1 7988   + caddc 7990  +∞cpnf 8166  ℤcz 9434  ♯chash 10984

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-ihash 10985

This theorem is referenced by:  hashcl  10990  hashfz1  10992  hashen  10993  fihashdom  11012  hashun  11014