Theorem hashennn 10751

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 10747	. . . . 5 ⊢ ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
2	1	fveq1i 5513	. . . 4 ⊢ (♯‘𝐴) = (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))‘𝐴)
3		funmpt 5251	. . . . 5 ⊢ Fun (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
4		hashennnuni 10750	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = 𝑁)
5	4	eqcomd 2183	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝑁 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
6		nnfi 6867	. . . . . . . . . . 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 6778	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝐴 ≈ 𝑁)
10		enfii 6869	. . . . . . . . . 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 4005	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → (𝑦 ≼ 𝑥 ↔ 𝑦 ≼ 𝐴))
15	14	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → (𝑦 ≼ 𝑥 ↔ 𝑦 ≼ 𝐴))
16	15	rabbidv 2726	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
17	16	unieqd 3819	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
18	13, 17	eqeq12d 2192	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → (𝑧 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} ↔ 𝑁 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}))
19	18	opelopabga 4261	. . . . . . . . 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 4098	. . . . . . 7 ⊢ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}}
23	21, 22	eleqtrrdi 2271	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ⟨𝐴, 𝑁⟩ ∈ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
24		opeldmg 4829	. . . . . . 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 5583	. . . . 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 2222	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴)))
30	11	elexd 2750	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝐴 ∈ V)
31	4, 12	eqeltrd 2254	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ∈ ω)
32	14	rabbidv 2726	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
33	32	unieqd 3819	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
34		eqid 2177	. . . . . . 7 ⊢ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) = (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
35	33, 34	fvmptg 5589	. . . . . 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 2210	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴) = 𝑁)
38	37	fveq2d 5516	. . 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 2210	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘𝑁))
40		ordom 4604	. . . . . . 7 ⊢ Ord ω
41		ordirr 4539	. . . . . . 7 ⊢ (Ord ω → ¬ ω ∈ ω)
42	40, 41	ax-mp 5	. . . . . 6 ⊢ ¬ ω ∈ ω
43		eleq1 2240	. . . . . 6 ⊢ (ω = 𝑁 → (ω ∈ ω ↔ 𝑁 ∈ ω))
44	42, 43	mtbii 674	. . . . 5 ⊢ (ω = 𝑁 → ¬ 𝑁 ∈ ω)
45	44	necon2ai 2401	. . . 4 ⊢ (𝑁 ∈ ω → ω ≠ 𝑁)
46		fvunsng 5707	. . . 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 2210	1 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (♯‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  {crab 2459  Vcvv 2737   ∪ cun 3127  {csn 3592  ⟨cop 3595  ∪ cuni 3808   class class class wbr 4001  {copab 4061   ↦ cmpt 4062  Ord word 4360  ωcom 4587  dom cdm 4624   ∘ ccom 4628  Fun wfun 5207  ‘cfv 5213  (class class class)co 5870  freccfrec 6386   ≈ cen 6733   ≼ cdom 6734  Fincfn 6735  0cc0 7806  1c1 7807   + caddc 7809  +∞cpnf 7983  ℤcz 9247  ♯chash 10746

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-iinf 4585

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-br 4002  df-opab 4063  df-mpt 4064  df-tr 4100  df-id 4291  df-iord 4364  df-on 4366  df-suc 4369  df-iom 4588  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-er 6530  df-en 6736  df-dom 6737  df-fin 6738  df-ihash 10747

This theorem is referenced by:  hashcl  10752  hashfz1  10754  hashen  10755  fihashdom  10774  hashun  10776