Theorem hashkf 14267

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 8376	. . . . . . 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 6597	. . . . . . 7 ⊢ (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω)
4	1, 3	mpbir 231	. . . . . 6 ⊢ 𝐺 Fn ω
5		fnfun 6600	. . . . . 6 ⊢ (𝐺 Fn ω → Fun 𝐺)
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun 𝐺
7		cardf2 9867	. . . . . 6 ⊢ card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On
8		ffun 6673	. . . . . 6 ⊢ (card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥 ≈ 𝑦}⟶On → Fun card)
9	7, 8	ax-mp 5	. . . . 5 ⊢ Fun card
10		funco 6540	. . . . 5 ⊢ ((Fun 𝐺 ∧ Fun card) → Fun (𝐺 ∘ card))
11	6, 9, 10	mp2an 693	. . . 4 ⊢ Fun (𝐺 ∘ card)
12		dmco 6221	. . . . 5 ⊢ dom (𝐺 ∘ card) = (◡card “ dom 𝐺)
13	4	fndmi 6604	. . . . . 6 ⊢ dom 𝐺 = ω
14	13	imaeq2i 6025	. . . . 5 ⊢ (◡card “ dom 𝐺) = (◡card “ ω)
15		funfn 6530	. . . . . . . . 9 ⊢ (Fun card ↔ card Fn dom card)
16	9, 15	mpbi 230	. . . . . . . 8 ⊢ card Fn dom card
17		elpreima 7012	. . . . . . . 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 9877	. . . . . . . . . . 11 ⊢ (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
21	20	ensymd 8954	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom card → 𝑦 ≈ (card‘𝑦))
22		breq2 5104	. . . . . . . . . . 11 ⊢ (𝑥 = (card‘𝑦) → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ (card‘𝑦)))
23	22	rspcev 3578	. . . . . . . . . 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 8924	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin ↔ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥)
26	24, 25	sylibr 234	. . . . . . . 8 ⊢ ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
27		finnum 9872	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → 𝑦 ∈ dom card)
28		ficardom 9885	. . . . . . . . 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 6503	. . . 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 6597	. . 3 ⊢ (𝐾 Fn Fin ↔ (𝐺 ∘ card) Fn Fin)
38	35, 37	mpbir 231	. 2 ⊢ 𝐾 Fn Fin
39	36	fveq1i 6843	. . . . 5 ⊢ (𝐾‘𝑦) = ((𝐺 ∘ card)‘𝑦)
40		fvco 6940	. . . . . 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 13906	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→ℕ0
44		f1of 6782	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ0 → 𝐺:ω⟶ℕ0)
45	43, 44	ax-mp 5	. . . . . 6 ⊢ 𝐺:ω⟶ℕ0
46	45	ffvelcdmi 7037	. . . . 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 7073	. 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 3442   class class class wbr 5100   ↦ cmpt 5181  ^◡ccnv 5631  dom cdm 5632   ↾ cres 5634   “ cima 5635   ∘ ccom 5636  Oncon0 6325  Fun wfun 6494   Fn wfn 6495  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368  ωcom 7818  reccrdg 8350   ≈ cen 8892  Fincfn 8895  cardccrd 9859  0cc0 11038  1c1 11039   + caddc 11041  ℕ₀cn0 12413

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 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-n0 12414  df-z 12501  df-uz 12764

This theorem is referenced by:  hashgval  14268  hashinf  14270  hashfxnn0  14272