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Theorem hashkf 13690
 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 8055 . . . . . . 7 (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω
2 hashgval.1 . . . . . . . 8 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
32fneq1i 6420 . . . . . . 7 (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω)
41, 3mpbir 234 . . . . . 6 𝐺 Fn ω
5 fnfun 6423 . . . . . 6 (𝐺 Fn ω → Fun 𝐺)
64, 5ax-mp 5 . . . . 5 Fun 𝐺
7 cardf2 9358 . . . . . 6 card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On
8 ffun 6490 . . . . . 6 (card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On → Fun card)
97, 8ax-mp 5 . . . . 5 Fun card
10 funco 6364 . . . . 5 ((Fun 𝐺 ∧ Fun card) → Fun (𝐺 ∘ card))
116, 9, 10mp2an 691 . . . 4 Fun (𝐺 ∘ card)
12 dmco 6074 . . . . 5 dom (𝐺 ∘ card) = (card “ dom 𝐺)
134fndmi 6426 . . . . . 6 dom 𝐺 = ω
1413imaeq2i 5894 . . . . 5 (card “ dom 𝐺) = (card “ ω)
15 funfn 6354 . . . . . . . . 9 (Fun card ↔ card Fn dom card)
169, 15mpbi 233 . . . . . . . 8 card Fn dom card
17 elpreima 6805 . . . . . . . 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 9368 . . . . . . . . . . 11 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
2120ensymd 8545 . . . . . . . . . 10 (𝑦 ∈ dom card → 𝑦 ≈ (card‘𝑦))
22 breq2 5034 . . . . . . . . . . 11 (𝑥 = (card‘𝑦) → (𝑦𝑥𝑦 ≈ (card‘𝑦)))
2322rspcev 3571 . . . . . . . . . 10 (((card‘𝑦) ∈ ω ∧ 𝑦 ≈ (card‘𝑦)) → ∃𝑥 ∈ ω 𝑦𝑥)
2419, 21, 23syl2anr 599 . . . . . . . . 9 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → ∃𝑥 ∈ ω 𝑦𝑥)
25 isfi 8518 . . . . . . . . 9 (𝑦 ∈ Fin ↔ ∃𝑥 ∈ ω 𝑦𝑥)
2624, 25sylibr 237 . . . . . . . 8 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
27 finnum 9363 . . . . . . . . 9 (𝑦 ∈ Fin → 𝑦 ∈ dom card)
28 ficardom 9376 . . . . . . . . 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 2795 . . . . 5 (card “ ω) = Fin
3312, 14, 323eqtri 2825 . . . 4 dom (𝐺 ∘ card) = Fin
34 df-fn 6327 . . . 4 ((𝐺 ∘ card) Fn Fin ↔ (Fun (𝐺 ∘ card) ∧ dom (𝐺 ∘ card) = Fin))
3511, 33, 34mpbir2an 710 . . 3 (𝐺 ∘ card) Fn Fin
36 hashkf.2 . . . 4 𝐾 = (𝐺 ∘ card)
3736fneq1i 6420 . . 3 (𝐾 Fn Fin ↔ (𝐺 ∘ card) Fn Fin)
3835, 37mpbir 234 . 2 𝐾 Fn Fin
3936fveq1i 6646 . . . . 5 (𝐾𝑦) = ((𝐺 ∘ card)‘𝑦)
40 fvco 6736 . . . . . 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, 41syl5eq 2845 . . . 4 (𝑦 ∈ Fin → (𝐾𝑦) = (𝐺‘(card‘𝑦)))
432hashgf1o 13336 . . . . . . 7 𝐺:ω–1-1-onto→ℕ0
44 f1of 6590 . . . . . . 7 (𝐺:ω–1-1-onto→ℕ0𝐺:ω⟶ℕ0)
4543, 44ax-mp 5 . . . . . 6 𝐺:ω⟶ℕ0
4645ffvelrni 6827 . . . . 5 ((card‘𝑦) ∈ ω → (𝐺‘(card‘𝑦)) ∈ ℕ0)
4728, 46syl 17 . . . 4 (𝑦 ∈ Fin → (𝐺‘(card‘𝑦)) ∈ ℕ0)
4842, 47eqeltrd 2890 . . 3 (𝑦 ∈ Fin → (𝐾𝑦) ∈ ℕ0)
4948rgen 3116 . 2 𝑦 ∈ Fin (𝐾𝑦) ∈ ℕ0
50 ffnfv 6859 . 2 (𝐾:Fin⟶ℕ0 ↔ (𝐾 Fn Fin ∧ ∀𝑦 ∈ Fin (𝐾𝑦) ∈ ℕ0))
5138, 49, 50mpbir2an 710 1 𝐾:Fin⟶ℕ0
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  Vcvv 3441   class class class wbr 5030   ↦ cmpt 5110  ◡ccnv 5518  dom cdm 5519   ↾ cres 5521   “ cima 5522   ∘ ccom 5523  Oncon0 6159  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135  ωcom 7562  reccrdg 8030   ≈ cen 8491  Fincfn 8494  cardccrd 9350  0cc0 10528  1c1 10529   + caddc 10531  ℕ0cn0 11887 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-card 9354  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-nn 11628  df-n0 11888  df-z 11972  df-uz 12234 This theorem is referenced by:  hashgval  13691  hashinf  13693  hashfxnn0  13695
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