Theorem hashkf 13546

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 7929	. . . . . . 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 6327	. . . . . . 7 ⊢ (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω)
4	1, 3	mpbir 232	. . . . . 6 ⊢ 𝐺 Fn ω
5		fnfun 6330	. . . . . 6 ⊢ (𝐺 Fn ω → Fun 𝐺)
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun 𝐺
7		cardf2 9225	. . . . . 6 ⊢ card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On
8		ffun 6392	. . . . . 6 ⊢ (card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On → Fun card)
9	7, 8	ax-mp 5	. . . . 5 ⊢ Fun card
10		funco 6272	. . . . 5 ⊢ ((Fun 𝐺 ∧ Fun card) → Fun (𝐺 ∘ card))
11	6, 9, 10	mp2an 688	. . . 4 ⊢ Fun (𝐺 ∘ card)
12		dmco 5989	. . . . 5 ⊢ dom (𝐺 ∘ card) = (◡card “ dom 𝐺)
13		fndm 6332	. . . . . . 7 ⊢ (𝐺 Fn ω → dom 𝐺 = ω)
14	4, 13	ax-mp 5	. . . . . 6 ⊢ dom 𝐺 = ω
15	14	imaeq2i 5811	. . . . 5 ⊢ (◡card “ dom 𝐺) = (◡card “ ω)
16		funfn 6262	. . . . . . . . 9 ⊢ (Fun card ↔ card Fn dom card)
17	9, 16	mpbi 231	. . . . . . . 8 ⊢ card Fn dom card
18		elpreima 6700	. . . . . . . 8 ⊢ (card Fn dom card → (𝑦 ∈ (◡card “ ω) ↔ (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω)))
19	17, 18	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ (◡card “ ω) ↔ (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω))
20		id 22	. . . . . . . . . 10 ⊢ ((card‘𝑦) ∈ ω → (card‘𝑦) ∈ ω)
21		cardid2 9235	. . . . . . . . . . 11 ⊢ (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
22	21	ensymd 8415	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom card → 𝑦 ≈ (card‘𝑦))
23		breq2 4972	. . . . . . . . . . 11 ⊢ (𝑥 = (card‘𝑦) → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ (card‘𝑦)))
24	23	rspcev 3561	. . . . . . . . . 10 ⊢ (((card‘𝑦) ∈ ω ∧ 𝑦 ≈ (card‘𝑦)) → ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
25	20, 22, 24	syl2anr 596	. . . . . . . . 9 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
26		isfi 8388	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin ↔ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
27	25, 26	sylibr 235	. . . . . . . 8 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
28		finnum 9230	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → 𝑦 ∈ dom card)
29		ficardom 9243	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
30	28, 29	jca 512	. . . . . . . 8 ⊢ (𝑦 ∈ Fin → (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω))
31	27, 30	impbii 210	. . . . . . 7 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) ↔ 𝑦 ∈ Fin)
32	19, 31	bitri 276	. . . . . 6 ⊢ (𝑦 ∈ (◡card “ ω) ↔ 𝑦 ∈ Fin)
33	32	eqriv 2794	. . . . 5 ⊢ (◡card “ ω) = Fin
34	12, 15, 33	3eqtri 2825	. . . 4 ⊢ dom (𝐺 ∘ card) = Fin
35		df-fn 6235	. . . 4 ⊢ ((𝐺 ∘ card) Fn Fin ↔ (Fun (𝐺 ∘ card) ∧ dom (𝐺 ∘ card) = Fin))
36	11, 34, 35	mpbir2an 707	. . 3 ⊢ (𝐺 ∘ card) Fn Fin
37		hashkf.2	. . . 4 ⊢ 𝐾 = (𝐺 ∘ card)
38	37	fneq1i 6327	. . 3 ⊢ (𝐾 Fn Fin ↔ (𝐺 ∘ card) Fn Fin)
39	36, 38	mpbir 232	. 2 ⊢ 𝐾 Fn Fin
40	37	fveq1i 6546	. . . . 5 ⊢ (𝐾‘𝑦) = ((𝐺 ∘ card)‘𝑦)
41		fvco 6633	. . . . . 6 ⊢ ((Fun card ∧ 𝑦 ∈ dom card) → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
42	9, 28, 41	sylancr 587	. . . . 5 ⊢ (𝑦 ∈ Fin → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
43	40, 42	syl5eq 2845	. . . 4 ⊢ (𝑦 ∈ Fin → (𝐾‘𝑦) = (𝐺‘(card‘𝑦)))
44	2	hashgf1o 13193	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→ℕ0
45		f1of 6490	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ0 → 𝐺:ω⟶ℕ0)
46	44, 45	ax-mp 5	. . . . . 6 ⊢ 𝐺:ω⟶ℕ0
47	46	ffvelrni 6722	. . . . 5 ⊢ ((card‘𝑦) ∈ ω → (𝐺‘(card‘𝑦)) ∈ ℕ0)
48	29, 47	syl 17	. . . 4 ⊢ (𝑦 ∈ Fin → (𝐺‘(card‘𝑦)) ∈ ℕ0)
49	43, 48	eqeltrd 2885	. . 3 ⊢ (𝑦 ∈ Fin → (𝐾‘𝑦) ∈ ℕ0)
50	49	rgen 3117	. 2 ⊢ ∀𝑦 ∈ Fin (𝐾‘𝑦) ∈ ℕ0
51		ffnfv 6752	. 2 ⊢ (𝐾:Fin⟶ℕ0 ↔ (𝐾 Fn Fin ∧ ∀𝑦 ∈ Fin (𝐾‘𝑦) ∈ ℕ0))
52	39, 50, 51	mpbir2an 707	1 ⊢ 𝐾:Fin⟶ℕ0

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1525   ∈ wcel 2083  {cab 2777  ∀wral 3107  ∃wrex 3108  Vcvv 3440   class class class wbr 4968   ↦ cmpt 5047  ^◡ccnv 5449  dom cdm 5450   ↾ cres 5452   “ cima 5453   ∘ ccom 5454  Oncon0 6073  Fun wfun 6226   Fn wfn 6227  ⟶wf 6228  –1-1-onto→wf1o 6231  ‘cfv 6232  (class class class)co 7023  ωcom 7443  reccrdg 7904   ≈ cen 8361  Fincfn 8364  cardccrd 9217  0cc0 10390  1c1 10391   + caddc 10393  ℕ₀cn0 11751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-n0 11752  df-z 11836  df-uz 12098

This theorem is referenced by:  hashgval  13547  hashinf  13549  hashfxnn0  13551