Theorem hashkf 14046

Description: The finite part of the size function maps all finite sets to their cardinality, as members of ℕ₀. (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)

Hypotheses

Ref	Expression
hashgval.1	⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
hashkf.2	⊢ 𝐾 = (𝐺 ∘ card)

Assertion

Ref	Expression
hashkf	⊢ 𝐾:Fin⟶ℕ0

Proof of Theorem hashkf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frfnom 8266	. . . . . . 7 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω
2		hashgval.1	. . . . . . . 8 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
3	2	fneq1i 6530	. . . . . . 7 ⊢ (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω)
4	1, 3	mpbir 230	. . . . . 6 ⊢ 𝐺 Fn ω
5		fnfun 6533	. . . . . 6 ⊢ (𝐺 Fn ω → Fun 𝐺)
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun 𝐺
7		cardf2 9701	. . . . . 6 ⊢ card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On
8		ffun 6603	. . . . . 6 ⊢ (card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On → Fun card)
9	7, 8	ax-mp 5	. . . . 5 ⊢ Fun card
10		funco 6474	. . . . 5 ⊢ ((Fun 𝐺 ∧ Fun card) → Fun (𝐺 ∘ card))
11	6, 9, 10	mp2an 689	. . . 4 ⊢ Fun (𝐺 ∘ card)
12		dmco 6158	. . . . 5 ⊢ dom (𝐺 ∘ card) = (◡card “ dom 𝐺)
13	4	fndmi 6537	. . . . . 6 ⊢ dom 𝐺 = ω
14	13	imaeq2i 5967	. . . . 5 ⊢ (◡card “ dom 𝐺) = (◡card “ ω)
15		funfn 6464	. . . . . . . . 9 ⊢ (Fun card ↔ card Fn dom card)
16	9, 15	mpbi 229	. . . . . . . 8 ⊢ card Fn dom card
17		elpreima 6935	. . . . . . . 8 ⊢ (card Fn dom card → (𝑦 ∈ (◡card “ ω) ↔ (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω)))
18	16, 17	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ (◡card “ ω) ↔ (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω))
19		id 22	. . . . . . . . . 10 ⊢ ((card‘𝑦) ∈ ω → (card‘𝑦) ∈ ω)
20		cardid2 9711	. . . . . . . . . . 11 ⊢ (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
21	20	ensymd 8791	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom card → 𝑦 ≈ (card‘𝑦))
22		breq2 5078	. . . . . . . . . . 11 ⊢ (𝑥 = (card‘𝑦) → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ (card‘𝑦)))
23	22	rspcev 3561	. . . . . . . . . 10 ⊢ (((card‘𝑦) ∈ ω ∧ 𝑦 ≈ (card‘𝑦)) → ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
24	19, 21, 23	syl2anr 597	. . . . . . . . 9 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
25		isfi 8764	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin ↔ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
26	24, 25	sylibr 233	. . . . . . . 8 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
27		finnum 9706	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → 𝑦 ∈ dom card)
28		ficardom 9719	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
29	27, 28	jca 512	. . . . . . . 8 ⊢ (𝑦 ∈ Fin → (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω))
30	26, 29	impbii 208	. . . . . . 7 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) ↔ 𝑦 ∈ Fin)
31	18, 30	bitri 274	. . . . . 6 ⊢ (𝑦 ∈ (◡card “ ω) ↔ 𝑦 ∈ Fin)
32	31	eqriv 2735	. . . . 5 ⊢ (◡card “ ω) = Fin
33	12, 14, 32	3eqtri 2770	. . . 4 ⊢ dom (𝐺 ∘ card) = Fin
34		df-fn 6436	. . . 4 ⊢ ((𝐺 ∘ card) Fn Fin ↔ (Fun (𝐺 ∘ card) ∧ dom (𝐺 ∘ card) = Fin))
35	11, 33, 34	mpbir2an 708	. . 3 ⊢ (𝐺 ∘ card) Fn Fin
36		hashkf.2	. . . 4 ⊢ 𝐾 = (𝐺 ∘ card)
37	36	fneq1i 6530	. . 3 ⊢ (𝐾 Fn Fin ↔ (𝐺 ∘ card) Fn Fin)
38	35, 37	mpbir 230	. 2 ⊢ 𝐾 Fn Fin
39	36	fveq1i 6775	. . . . 5 ⊢ (𝐾‘𝑦) = ((𝐺 ∘ card)‘𝑦)
40		fvco 6866	. . . . . 6 ⊢ ((Fun card ∧ 𝑦 ∈ dom card) → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
41	9, 27, 40	sylancr 587	. . . . 5 ⊢ (𝑦 ∈ Fin → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
42	39, 41	eqtrid 2790	. . . 4 ⊢ (𝑦 ∈ Fin → (𝐾‘𝑦) = (𝐺‘(card‘𝑦)))
43	2	hashgf1o 13691	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→ℕ0
44		f1of 6716	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ0 → 𝐺:ω⟶ℕ0)
45	43, 44	ax-mp 5	. . . . . 6 ⊢ 𝐺:ω⟶ℕ0
46	45	ffvelrni 6960	. . . . 5 ⊢ ((card‘𝑦) ∈ ω → (𝐺‘(card‘𝑦)) ∈ ℕ0)
47	28, 46	syl 17	. . . 4 ⊢ (𝑦 ∈ Fin → (𝐺‘(card‘𝑦)) ∈ ℕ0)
48	42, 47	eqeltrd 2839	. . 3 ⊢ (𝑦 ∈ Fin → (𝐾‘𝑦) ∈ ℕ0)
49	48	rgen 3074	. 2 ⊢ ∀𝑦 ∈ Fin (𝐾‘𝑦) ∈ ℕ0
50		ffnfv 6992	. 2 ⊢ (𝐾:Fin⟶ℕ0 ↔ (𝐾 Fn Fin ∧ ∀𝑦 ∈ Fin (𝐾‘𝑦) ∈ ℕ0))
51	38, 49, 50	mpbir2an 708	1 ⊢ 𝐾:Fin⟶ℕ0

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065  Vcvv 3432   class class class wbr 5074   ↦ cmpt 5157  ^◡ccnv 5588  dom cdm 5589   ↾ cres 5591   “ cima 5592   ∘ ccom 5593  Oncon0 6266  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275  ωcom 7712  reccrdg 8240   ≈ cen 8730  Fincfn 8733  cardccrd 9693  0cc0 10871  1c1 10872   + caddc 10874  ℕ₀cn0 12233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583

This theorem is referenced by:  hashgval  14047  hashinf  14049  hashfxnn0  14051