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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 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.
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-onto→wf1o 5925  ‘cfv 5926  (class class class)co 6690  ωcom 7107  reccrdg 7550   ≈ cen 7994  Fincfn 7997  cardccrd 8799  0cc0 9974  1c1 9975   + caddc 9977  ℕ0cn0 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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