Theorem hashennn 10889

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 10885	. . . . 5 ⊢ ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
2	1	fveq1i 5562	. . . 4 ⊢ (♯‘𝐴) = (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))‘𝐴)
3		funmpt 5297	. . . . 5 ⊢ Fun (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
4		hashennnuni 10888	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = 𝑁)
5	4	eqcomd 2202	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝑁 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
6		nnfi 6942	. . . . . . . . . . 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 6851	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝐴 ≈ 𝑁)
10		enfii 6944	. . . . . . . . . 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 4038	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → (𝑦 ≼ 𝑥 ↔ 𝑦 ≼ 𝐴))
15	14	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → (𝑦 ≼ 𝑥 ↔ 𝑦 ≼ 𝐴))
16	15	rabbidv 2752	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
17	16	unieqd 3851	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
18	13, 17	eqeq12d 2211	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑁) → (𝑧 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} ↔ 𝑁 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}))
19	18	opelopabga 4298	. . . . . . . . 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 4131	. . . . . . 7 ⊢ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}}
23	21, 22	eleqtrrdi 2290	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ⟨𝐴, 𝑁⟩ ∈ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
24		opeldmg 4872	. . . . . . 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 5634	. . . . 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 2241	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴)))
30	11	elexd 2776	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → 𝐴 ∈ V)
31	4, 12	eqeltrd 2273	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ∈ ω)
32	14	rabbidv 2752	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
33	32	unieqd 3851	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥} = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
34		eqid 2196	. . . . . . 7 ⊢ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}) = (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
35	33, 34	fvmptg 5640	. . . . . 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 2229	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ((𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})‘𝐴) = 𝑁)
38	37	fveq2d 5565	. . 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 2229	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘𝑁))
40		ordom 4644	. . . . . . 7 ⊢ Ord ω
41		ordirr 4579	. . . . . . 7 ⊢ (Ord ω → ¬ ω ∈ ω)
42	40, 41	ax-mp 5	. . . . . 6 ⊢ ¬ ω ∈ ω
43		eleq1 2259	. . . . . 6 ⊢ (ω = 𝑁 → (ω ∈ ω ↔ 𝑁 ∈ ω))
44	42, 43	mtbii 675	. . . . 5 ⊢ (ω = 𝑁 → ¬ 𝑁 ∈ ω)
45	44	necon2ai 2421	. . . 4 ⊢ (𝑁 ∈ ω → ω ≠ 𝑁)
46		fvunsng 5759	. . . 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 2229	1 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (♯‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  {crab 2479  Vcvv 2763   ∪ cun 3155  {csn 3623  ⟨cop 3626  ∪ cuni 3840   class class class wbr 4034  {copab 4094   ↦ cmpt 4095  Ord word 4398  ωcom 4627  dom cdm 4664   ∘ ccom 4668  Fun wfun 5253  ‘cfv 5259  (class class class)co 5925  freccfrec 6457   ≈ cen 6806   ≼ cdom 6807  Fincfn 6808  0cc0 7896  1c1 7897   + caddc 7899  +∞cpnf 8075  ℤcz 9343  ♯chash 10884

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-ihash 10885

This theorem is referenced by:  hashcl  10890  hashfz1  10892  hashen  10893  fihashdom  10912  hashun  10914