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Theorem hashkf 14371
Description: The finite part of the size function maps all finite sets to their cardinality, as members of 0. (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.
StepHypRef Expression
1 frfnom 8475 . . . . . . 7 (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω
2 hashgval.1 . . . . . . . 8 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
32fneq1i 6665 . . . . . . 7 (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω)
41, 3mpbir 231 . . . . . 6 𝐺 Fn ω
5 fnfun 6668 . . . . . 6 (𝐺 Fn ω → Fun 𝐺)
64, 5ax-mp 5 . . . . 5 Fun 𝐺
7 cardf2 9983 . . . . . 6 card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On
8 ffun 6739 . . . . . 6 (card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On → Fun card)
97, 8ax-mp 5 . . . . 5 Fun card
10 funco 6606 . . . . 5 ((Fun 𝐺 ∧ Fun card) → Fun (𝐺 ∘ card))
116, 9, 10mp2an 692 . . . 4 Fun (𝐺 ∘ card)
12 dmco 6274 . . . . 5 dom (𝐺 ∘ card) = (card “ dom 𝐺)
134fndmi 6672 . . . . . 6 dom 𝐺 = ω
1413imaeq2i 6076 . . . . 5 (card “ dom 𝐺) = (card “ ω)
15 funfn 6596 . . . . . . . . 9 (Fun card ↔ card Fn dom card)
169, 15mpbi 230 . . . . . . . 8 card Fn dom card
17 elpreima 7078 . . . . . . . 8 (card Fn dom card → (𝑦 ∈ (card “ ω) ↔ (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω)))
1816, 17ax-mp 5 . . . . . . 7 (𝑦 ∈ (card “ ω) ↔ (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω))
19 id 22 . . . . . . . . . 10 ((card‘𝑦) ∈ ω → (card‘𝑦) ∈ ω)
20 cardid2 9993 . . . . . . . . . . 11 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
2120ensymd 9045 . . . . . . . . . 10 (𝑦 ∈ dom card → 𝑦 ≈ (card‘𝑦))
22 breq2 5147 . . . . . . . . . . 11 (𝑥 = (card‘𝑦) → (𝑦𝑥𝑦 ≈ (card‘𝑦)))
2322rspcev 3622 . . . . . . . . . 10 (((card‘𝑦) ∈ ω ∧ 𝑦 ≈ (card‘𝑦)) → ∃𝑥 ∈ ω 𝑦𝑥)
2419, 21, 23syl2anr 597 . . . . . . . . 9 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → ∃𝑥 ∈ ω 𝑦𝑥)
25 isfi 9016 . . . . . . . . 9 (𝑦 ∈ Fin ↔ ∃𝑥 ∈ ω 𝑦𝑥)
2624, 25sylibr 234 . . . . . . . 8 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
27 finnum 9988 . . . . . . . . 9 (𝑦 ∈ Fin → 𝑦 ∈ dom card)
28 ficardom 10001 . . . . . . . . 9 (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
2927, 28jca 511 . . . . . . . 8 (𝑦 ∈ Fin → (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω))
3026, 29impbii 209 . . . . . . 7 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) ↔ 𝑦 ∈ Fin)
3118, 30bitri 275 . . . . . 6 (𝑦 ∈ (card “ ω) ↔ 𝑦 ∈ Fin)
3231eqriv 2734 . . . . 5 (card “ ω) = Fin
3312, 14, 323eqtri 2769 . . . 4 dom (𝐺 ∘ card) = Fin
34 df-fn 6564 . . . 4 ((𝐺 ∘ card) Fn Fin ↔ (Fun (𝐺 ∘ card) ∧ dom (𝐺 ∘ card) = Fin))
3511, 33, 34mpbir2an 711 . . 3 (𝐺 ∘ card) Fn Fin
36 hashkf.2 . . . 4 𝐾 = (𝐺 ∘ card)
3736fneq1i 6665 . . 3 (𝐾 Fn Fin ↔ (𝐺 ∘ card) Fn Fin)
3835, 37mpbir 231 . 2 𝐾 Fn Fin
3936fveq1i 6907 . . . . 5 (𝐾𝑦) = ((𝐺 ∘ card)‘𝑦)
40 fvco 7007 . . . . . 6 ((Fun card ∧ 𝑦 ∈ dom card) → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
419, 27, 40sylancr 587 . . . . 5 (𝑦 ∈ Fin → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
4239, 41eqtrid 2789 . . . 4 (𝑦 ∈ Fin → (𝐾𝑦) = (𝐺‘(card‘𝑦)))
432hashgf1o 14012 . . . . . . 7 𝐺:ω–1-1-onto→ℕ0
44 f1of 6848 . . . . . . 7 (𝐺:ω–1-1-onto→ℕ0𝐺:ω⟶ℕ0)
4543, 44ax-mp 5 . . . . . 6 𝐺:ω⟶ℕ0
4645ffvelcdmi 7103 . . . . 5 ((card‘𝑦) ∈ ω → (𝐺‘(card‘𝑦)) ∈ ℕ0)
4728, 46syl 17 . . . 4 (𝑦 ∈ Fin → (𝐺‘(card‘𝑦)) ∈ ℕ0)
4842, 47eqeltrd 2841 . . 3 (𝑦 ∈ Fin → (𝐾𝑦) ∈ ℕ0)
4948rgen 3063 . 2 𝑦 ∈ Fin (𝐾𝑦) ∈ ℕ0
50 ffnfv 7139 . 2 (𝐾:Fin⟶ℕ0 ↔ (𝐾 Fn Fin ∧ ∀𝑦 ∈ Fin (𝐾𝑦) ∈ ℕ0))
5138, 49, 50mpbir2an 711 1 𝐾:Fin⟶ℕ0
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480   class class class wbr 5143  cmpt 5225  ccnv 5684  dom cdm 5685  cres 5687  cima 5688  ccom 5689  Oncon0 6384  Fun wfun 6555   Fn wfn 6556  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  ωcom 7887  reccrdg 8449  cen 8982  Fincfn 8985  cardccrd 9975  0cc0 11155  1c1 11156   + caddc 11158  0cn0 12526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879
This theorem is referenced by:  hashgval  14372  hashinf  14374  hashfxnn0  14376
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