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Theorem hashkf 13898
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 8170 . . . . . . 7 (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω
2 hashgval.1 . . . . . . . 8 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
32fneq1i 6476 . . . . . . 7 (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω)
41, 3mpbir 234 . . . . . 6 𝐺 Fn ω
5 fnfun 6479 . . . . . 6 (𝐺 Fn ω → Fun 𝐺)
64, 5ax-mp 5 . . . . 5 Fun 𝐺
7 cardf2 9559 . . . . . 6 card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On
8 ffun 6548 . . . . . 6 (card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On → Fun card)
97, 8ax-mp 5 . . . . 5 Fun card
10 funco 6420 . . . . 5 ((Fun 𝐺 ∧ Fun card) → Fun (𝐺 ∘ card))
116, 9, 10mp2an 692 . . . 4 Fun (𝐺 ∘ card)
12 dmco 6118 . . . . 5 dom (𝐺 ∘ card) = (card “ dom 𝐺)
134fndmi 6482 . . . . . 6 dom 𝐺 = ω
1413imaeq2i 5927 . . . . 5 (card “ dom 𝐺) = (card “ ω)
15 funfn 6410 . . . . . . . . 9 (Fun card ↔ card Fn dom card)
169, 15mpbi 233 . . . . . . . 8 card Fn dom card
17 elpreima 6878 . . . . . . . 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 9569 . . . . . . . . . . 11 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
2120ensymd 8679 . . . . . . . . . 10 (𝑦 ∈ dom card → 𝑦 ≈ (card‘𝑦))
22 breq2 5057 . . . . . . . . . . 11 (𝑥 = (card‘𝑦) → (𝑦𝑥𝑦 ≈ (card‘𝑦)))
2322rspcev 3537 . . . . . . . . . 10 (((card‘𝑦) ∈ ω ∧ 𝑦 ≈ (card‘𝑦)) → ∃𝑥 ∈ ω 𝑦𝑥)
2419, 21, 23syl2anr 600 . . . . . . . . 9 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → ∃𝑥 ∈ ω 𝑦𝑥)
25 isfi 8652 . . . . . . . . 9 (𝑦 ∈ Fin ↔ ∃𝑥 ∈ ω 𝑦𝑥)
2624, 25sylibr 237 . . . . . . . 8 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
27 finnum 9564 . . . . . . . . 9 (𝑦 ∈ Fin → 𝑦 ∈ dom card)
28 ficardom 9577 . . . . . . . . 9 (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
2927, 28jca 515 . . . . . . . 8 (𝑦 ∈ Fin → (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω))
3026, 29impbii 212 . . . . . . 7 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) ↔ 𝑦 ∈ Fin)
3118, 30bitri 278 . . . . . 6 (𝑦 ∈ (card “ ω) ↔ 𝑦 ∈ Fin)
3231eqriv 2734 . . . . 5 (card “ ω) = Fin
3312, 14, 323eqtri 2769 . . . 4 dom (𝐺 ∘ card) = Fin
34 df-fn 6383 . . . 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 6476 . . 3 (𝐾 Fn Fin ↔ (𝐺 ∘ card) Fn Fin)
3835, 37mpbir 234 . 2 𝐾 Fn Fin
3936fveq1i 6718 . . . . 5 (𝐾𝑦) = ((𝐺 ∘ card)‘𝑦)
40 fvco 6809 . . . . . 6 ((Fun card ∧ 𝑦 ∈ dom card) → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
419, 27, 40sylancr 590 . . . . 5 (𝑦 ∈ Fin → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
4239, 41eqtrid 2789 . . . 4 (𝑦 ∈ Fin → (𝐾𝑦) = (𝐺‘(card‘𝑦)))
432hashgf1o 13544 . . . . . . 7 𝐺:ω–1-1-onto→ℕ0
44 f1of 6661 . . . . . . 7 (𝐺:ω–1-1-onto→ℕ0𝐺:ω⟶ℕ0)
4543, 44ax-mp 5 . . . . . 6 𝐺:ω⟶ℕ0
4645ffvelrni 6903 . . . . 5 ((card‘𝑦) ∈ ω → (𝐺‘(card‘𝑦)) ∈ ℕ0)
4728, 46syl 17 . . . 4 (𝑦 ∈ Fin → (𝐺‘(card‘𝑦)) ∈ ℕ0)
4842, 47eqeltrd 2838 . . 3 (𝑦 ∈ Fin → (𝐾𝑦) ∈ ℕ0)
4948rgen 3071 . 2 𝑦 ∈ Fin (𝐾𝑦) ∈ ℕ0
50 ffnfv 6935 . 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 209  wa 399   = wceq 1543  wcel 2110  {cab 2714  wral 3061  wrex 3062  Vcvv 3408   class class class wbr 5053  cmpt 5135  ccnv 5550  dom cdm 5551  cres 5553  cima 5554  ccom 5555  Oncon0 6213  Fun wfun 6374   Fn wfn 6375  wf 6376  1-1-ontowf1o 6379  cfv 6380  (class class class)co 7213  ωcom 7644  reccrdg 8145  cen 8623  Fincfn 8626  cardccrd 9551  0cc0 10729  1c1 10730   + caddc 10732  0cn0 12090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-n0 12091  df-z 12177  df-uz 12439
This theorem is referenced by:  hashgval  13899  hashinf  13901  hashfxnn0  13903
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