Theorem hashkf 14285

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 8367	. . . . . . 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 6589	. . . . . . 7 ⊢ (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω)
4	1, 3	mpbir 231	. . . . . 6 ⊢ 𝐺 Fn ω
5		fnfun 6592	. . . . . 6 ⊢ (𝐺 Fn ω → Fun 𝐺)
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun 𝐺
7		cardf2 9858	. . . . . 6 ⊢ card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On
8		ffun 6665	. . . . . 6 ⊢ (card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On → Fun card)
9	7, 8	ax-mp 5	. . . . 5 ⊢ Fun card
10		funco 6532	. . . . 5 ⊢ ((Fun 𝐺 ∧ Fun card) → Fun (𝐺 ∘ card))
11	6, 9, 10	mp2an 693	. . . 4 ⊢ Fun (𝐺 ∘ card)
12		dmco 6213	. . . . 5 ⊢ dom (𝐺 ∘ card) = (◡card “ dom 𝐺)
13	4	fndmi 6596	. . . . . 6 ⊢ dom 𝐺 = ω
14	13	imaeq2i 6017	. . . . 5 ⊢ (◡card “ dom 𝐺) = (◡card “ ω)
15		funfn 6522	. . . . . . . . 9 ⊢ (Fun card ↔ card Fn dom card)
16	9, 15	mpbi 230	. . . . . . . 8 ⊢ card Fn dom card
17		elpreima 7004	. . . . . . . 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 9868	. . . . . . . . . . 11 ⊢ (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
21	20	ensymd 8945	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom card → 𝑦 ≈ (card‘𝑦))
22		breq2 5090	. . . . . . . . . . 11 ⊢ (𝑥 = (card‘𝑦) → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ (card‘𝑦)))
23	22	rspcev 3565	. . . . . . . . . 10 ⊢ (((card‘𝑦) ∈ ω ∧ 𝑦 ≈ (card‘𝑦)) → ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
24	19, 21, 23	syl2anr 598	. . . . . . . . 9 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
25		isfi 8915	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin ↔ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
26	24, 25	sylibr 234	. . . . . . . 8 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
27		finnum 9863	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → 𝑦 ∈ dom card)
28		ficardom 9876	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
29	27, 28	jca 511	. . . . . . . 8 ⊢ (𝑦 ∈ Fin → (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω))
30	26, 29	impbii 209	. . . . . . 7 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) ↔ 𝑦 ∈ Fin)
31	18, 30	bitri 275	. . . . . 6 ⊢ (𝑦 ∈ (◡card “ ω) ↔ 𝑦 ∈ Fin)
32	31	eqriv 2734	. . . . 5 ⊢ (◡card “ ω) = Fin
33	12, 14, 32	3eqtri 2764	. . . 4 ⊢ dom (𝐺 ∘ card) = Fin
34		df-fn 6495	. . . 4 ⊢ ((𝐺 ∘ card) Fn Fin ↔ (Fun (𝐺 ∘ card) ∧ dom (𝐺 ∘ card) = Fin))
35	11, 33, 34	mpbir2an 712	. . 3 ⊢ (𝐺 ∘ card) Fn Fin
36		hashkf.2	. . . 4 ⊢ 𝐾 = (𝐺 ∘ card)
37	36	fneq1i 6589	. . 3 ⊢ (𝐾 Fn Fin ↔ (𝐺 ∘ card) Fn Fin)
38	35, 37	mpbir 231	. 2 ⊢ 𝐾 Fn Fin
39	36	fveq1i 6835	. . . . 5 ⊢ (𝐾‘𝑦) = ((𝐺 ∘ card)‘𝑦)
40		fvco 6932	. . . . . 6 ⊢ ((Fun card ∧ 𝑦 ∈ dom card) → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
41	9, 27, 40	sylancr 588	. . . . 5 ⊢ (𝑦 ∈ Fin → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
42	39, 41	eqtrid 2784	. . . 4 ⊢ (𝑦 ∈ Fin → (𝐾‘𝑦) = (𝐺‘(card‘𝑦)))
43	2	hashgf1o 13924	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→ℕ0
44		f1of 6774	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ0 → 𝐺:ω⟶ℕ0)
45	43, 44	ax-mp 5	. . . . . 6 ⊢ 𝐺:ω⟶ℕ0
46	45	ffvelcdmi 7029	. . . . 5 ⊢ ((card‘𝑦) ∈ ω → (𝐺‘(card‘𝑦)) ∈ ℕ0)
47	28, 46	syl 17	. . . 4 ⊢ (𝑦 ∈ Fin → (𝐺‘(card‘𝑦)) ∈ ℕ0)
48	42, 47	eqeltrd 2837	. . 3 ⊢ (𝑦 ∈ Fin → (𝐾‘𝑦) ∈ ℕ0)
49	48	rgen 3054	. 2 ⊢ ∀𝑦 ∈ Fin (𝐾‘𝑦) ∈ ℕ0
50		ffnfv 7065	. 2 ⊢ (𝐾:Fin⟶ℕ0 ↔ (𝐾 Fn Fin ∧ ∀𝑦 ∈ Fin (𝐾‘𝑦) ∈ ℕ0))
51	38, 49, 50	mpbir2an 712	1 ⊢ 𝐾:Fin⟶ℕ0

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  Vcvv 3430   class class class wbr 5086   ↦ cmpt 5167  ^◡ccnv 5623  dom cdm 5624   ↾ cres 5626   “ cima 5627   ∘ ccom 5628  Oncon0 6317  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7360  ωcom 7810  reccrdg 8341   ≈ cen 8883  Fincfn 8886  cardccrd 9850  0cc0 11029  1c1 11030   + caddc 11032  ℕ₀cn0 12428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780

This theorem is referenced by:  hashgval  14286  hashinf  14288  hashfxnn0  14290