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Theorem hashkf 13159
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 7575 . . . . . . 7 (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω
2 hashgval.1 . . . . . . . 8 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
32fneq1i 6023 . . . . . . 7 (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω)
41, 3mpbir 221 . . . . . 6 𝐺 Fn ω
5 fnfun 6026 . . . . . 6 (𝐺 Fn ω → Fun 𝐺)
64, 5ax-mp 5 . . . . 5 Fun 𝐺
7 cardf2 8807 . . . . . 6 card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On
8 ffun 6086 . . . . . 6 (card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On → Fun card)
97, 8ax-mp 5 . . . . 5 Fun card
10 funco 5966 . . . . 5 ((Fun 𝐺 ∧ Fun card) → Fun (𝐺 ∘ card))
116, 9, 10mp2an 708 . . . 4 Fun (𝐺 ∘ card)
12 dmco 5681 . . . . 5 dom (𝐺 ∘ card) = (card “ dom 𝐺)
13 fndm 6028 . . . . . . 7 (𝐺 Fn ω → dom 𝐺 = ω)
144, 13ax-mp 5 . . . . . 6 dom 𝐺 = ω
1514imaeq2i 5499 . . . . 5 (card “ dom 𝐺) = (card “ ω)
16 funfn 5956 . . . . . . . . 9 (Fun card ↔ card Fn dom card)
179, 16mpbi 220 . . . . . . . 8 card Fn dom card
18 elpreima 6377 . . . . . . . 8 (card Fn dom card → (𝑦 ∈ (card “ ω) ↔ (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω)))
1917, 18ax-mp 5 . . . . . . 7 (𝑦 ∈ (card “ ω) ↔ (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω))
20 id 22 . . . . . . . . . 10 ((card‘𝑦) ∈ ω → (card‘𝑦) ∈ ω)
21 cardid2 8817 . . . . . . . . . . 11 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
2221ensymd 8048 . . . . . . . . . 10 (𝑦 ∈ dom card → 𝑦 ≈ (card‘𝑦))
23 breq2 4689 . . . . . . . . . . 11 (𝑥 = (card‘𝑦) → (𝑦𝑥𝑦 ≈ (card‘𝑦)))
2423rspcev 3340 . . . . . . . . . 10 (((card‘𝑦) ∈ ω ∧ 𝑦 ≈ (card‘𝑦)) → ∃𝑥 ∈ ω 𝑦𝑥)
2520, 22, 24syl2anr 494 . . . . . . . . 9 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → ∃𝑥 ∈ ω 𝑦𝑥)
26 isfi 8021 . . . . . . . . 9 (𝑦 ∈ Fin ↔ ∃𝑥 ∈ ω 𝑦𝑥)
2725, 26sylibr 224 . . . . . . . 8 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
28 finnum 8812 . . . . . . . . 9 (𝑦 ∈ Fin → 𝑦 ∈ dom card)
29 ficardom 8825 . . . . . . . . 9 (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
3028, 29jca 553 . . . . . . . 8 (𝑦 ∈ Fin → (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω))
3127, 30impbii 199 . . . . . . 7 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) ↔ 𝑦 ∈ Fin)
3219, 31bitri 264 . . . . . 6 (𝑦 ∈ (card “ ω) ↔ 𝑦 ∈ Fin)
3332eqriv 2648 . . . . 5 (card “ ω) = Fin
3412, 15, 333eqtri 2677 . . . 4 dom (𝐺 ∘ card) = Fin
35 df-fn 5929 . . . 4 ((𝐺 ∘ card) Fn Fin ↔ (Fun (𝐺 ∘ card) ∧ dom (𝐺 ∘ card) = Fin))
3611, 34, 35mpbir2an 975 . . 3 (𝐺 ∘ card) Fn Fin
37 hashkf.2 . . . 4 𝐾 = (𝐺 ∘ card)
3837fneq1i 6023 . . 3 (𝐾 Fn Fin ↔ (𝐺 ∘ card) Fn Fin)
3936, 38mpbir 221 . 2 𝐾 Fn Fin
4037fveq1i 6230 . . . . 5 (𝐾𝑦) = ((𝐺 ∘ card)‘𝑦)
41 fvco 6313 . . . . . 6 ((Fun card ∧ 𝑦 ∈ dom card) → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
429, 28, 41sylancr 696 . . . . 5 (𝑦 ∈ Fin → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
4340, 42syl5eq 2697 . . . 4 (𝑦 ∈ Fin → (𝐾𝑦) = (𝐺‘(card‘𝑦)))
442hashgf1o 12810 . . . . . . 7 𝐺:ω–1-1-onto→ℕ0
45 f1of 6175 . . . . . . 7 (𝐺:ω–1-1-onto→ℕ0𝐺:ω⟶ℕ0)
4644, 45ax-mp 5 . . . . . 6 𝐺:ω⟶ℕ0
4746ffvelrni 6398 . . . . 5 ((card‘𝑦) ∈ ω → (𝐺‘(card‘𝑦)) ∈ ℕ0)
4829, 47syl 17 . . . 4 (𝑦 ∈ Fin → (𝐺‘(card‘𝑦)) ∈ ℕ0)
4943, 48eqeltrd 2730 . . 3 (𝑦 ∈ Fin → (𝐾𝑦) ∈ ℕ0)
5049rgen 2951 . 2 𝑦 ∈ Fin (𝐾𝑦) ∈ ℕ0
51 ffnfv 6428 . 2 (𝐾:Fin⟶ℕ0 ↔ (𝐾 Fn Fin ∧ ∀𝑦 ∈ Fin (𝐾𝑦) ∈ ℕ0))
5239, 50, 51mpbir2an 975 1 𝐾:Fin⟶ℕ0
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231   class class class wbr 4685  cmpt 4762  ccnv 5142  dom cdm 5143  cres 5145  cima 5146  ccom 5147  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  ωcom 7107  reccrdg 7550  cen 7994  Fincfn 7997  cardccrd 8799  0cc0 9974  1c1 9975   + caddc 9977  0cn0 11330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726
This theorem is referenced by:  hashgval  13160  hashinf  13162  hashfxnn0  13164  hashfOLD  13166
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