Theorem hashkf 14292

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 8371	. . . . . . 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 232	. . . . . 6 ⊢ 𝐺 Fn ω
5		fnfun 6592	. . . . . 6 ⊢ (𝐺 Fn ω → Fun 𝐺)
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun 𝐺
7		cardf2 9865	. . . . . 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 698	. . . 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 231	. . . . . . . 8 ⊢ card Fn dom card
17		elpreima 7006	. . . . . . . 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 9875	. . . . . . . . . . 11 ⊢ (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
21	20	ensymd 8949	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom card → 𝑦 ≈ (card‘𝑦))
22		breq2 5083	. . . . . . . . . . 11 ⊢ (𝑥 = (card‘𝑦) → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ (card‘𝑦)))
23	22	rspcev 3567	. . . . . . . . . 10 ⊢ (((card‘𝑦) ∈ ω ∧ 𝑦 ≈ (card‘𝑦)) → ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
24	19, 21, 23	syl2anr 603	. . . . . . . . 9 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
25		isfi 8919	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin ↔ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
26	24, 25	sylibr 235	. . . . . . . 8 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
27		finnum 9870	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → 𝑦 ∈ dom card)
28		ficardom 9883	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
29	27, 28	jca 516	. . . . . . . 8 ⊢ (𝑦 ∈ Fin → (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω))
30	26, 29	impbii 210	. . . . . . 7 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) ↔ 𝑦 ∈ Fin)
31	18, 30	bitri 276	. . . . . 6 ⊢ (𝑦 ∈ (◡card “ ω) ↔ 𝑦 ∈ Fin)
32	31	eqriv 2737	. . . . 5 ⊢ (◡card “ ω) = Fin
33	12, 14, 32	3eqtri 2767	. . . 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 717	. . 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 232	. 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 593	. . . . 5 ⊢ (𝑦 ∈ Fin → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
42	39, 41	eqtrid 2787	. . . 4 ⊢ (𝑦 ∈ Fin → (𝐾‘𝑦) = (𝐺‘(card‘𝑦)))
43	2	hashgf1o 13931	. . . . . . 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 7031	. . . . 5 ⊢ ((card‘𝑦) ∈ ω → (𝐺‘(card‘𝑦)) ∈ ℕ0)
47	28, 46	syl 17	. . . 4 ⊢ (𝑦 ∈ Fin → (𝐺‘(card‘𝑦)) ∈ ℕ0)
48	42, 47	eqeltrd 2840	. . 3 ⊢ (𝑦 ∈ Fin → (𝐾‘𝑦) ∈ ℕ0)
49	48	rgen 3056	. 2 ⊢ ∀𝑦 ∈ Fin (𝐾‘𝑦) ∈ ℕ0
50		ffnfv 7067	. 2 ⊢ (𝐾:Fin⟶ℕ0 ↔ (𝐾 Fn Fin ∧ ∀𝑦 ∈ Fin (𝐾‘𝑦) ∈ ℕ0))
51	38, 49, 50	mpbir2an 717	1 ⊢ 𝐾:Fin⟶ℕ0

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2718  ∀wral 3054  ∃wrex 3064  Vcvv 3432   class class class wbr 5079   ↦ cmpt 5160  ^◡ccnv 5624  dom cdm 5625   ↾ cres 5627   “ cima 5628   ∘ ccom 5629  Oncon0 6317  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7363  ωcom 7813  reccrdg 8345   ≈ cen 8887  Fincfn 8890  cardccrd 9857  0cc0 11036  1c1 11037   + caddc 11039  ℕ₀cn0 12435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-n0 12436  df-z 12523  df-uz 12787

This theorem is referenced by:  hashgval  14293  hashinf  14295  hashfxnn0  14297