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Theorem hashkf 13546
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 7929 . . . . . . 7 (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω
2 hashgval.1 . . . . . . . 8 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
32fneq1i 6327 . . . . . . 7 (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) Fn ω)
41, 3mpbir 232 . . . . . 6 𝐺 Fn ω
5 fnfun 6330 . . . . . 6 (𝐺 Fn ω → Fun 𝐺)
64, 5ax-mp 5 . . . . 5 Fun 𝐺
7 cardf2 9225 . . . . . 6 card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On
8 ffun 6392 . . . . . 6 (card:{𝑦 ∣ ∃𝑥 ∈ On 𝑥𝑦}⟶On → Fun card)
97, 8ax-mp 5 . . . . 5 Fun card
10 funco 6272 . . . . 5 ((Fun 𝐺 ∧ Fun card) → Fun (𝐺 ∘ card))
116, 9, 10mp2an 688 . . . 4 Fun (𝐺 ∘ card)
12 dmco 5989 . . . . 5 dom (𝐺 ∘ card) = (card “ dom 𝐺)
13 fndm 6332 . . . . . . 7 (𝐺 Fn ω → dom 𝐺 = ω)
144, 13ax-mp 5 . . . . . 6 dom 𝐺 = ω
1514imaeq2i 5811 . . . . 5 (card “ dom 𝐺) = (card “ ω)
16 funfn 6262 . . . . . . . . 9 (Fun card ↔ card Fn dom card)
179, 16mpbi 231 . . . . . . . 8 card Fn dom card
18 elpreima 6700 . . . . . . . 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 9235 . . . . . . . . . . 11 (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
2221ensymd 8415 . . . . . . . . . 10 (𝑦 ∈ dom card → 𝑦 ≈ (card‘𝑦))
23 breq2 4972 . . . . . . . . . . 11 (𝑥 = (card‘𝑦) → (𝑦𝑥𝑦 ≈ (card‘𝑦)))
2423rspcev 3561 . . . . . . . . . 10 (((card‘𝑦) ∈ ω ∧ 𝑦 ≈ (card‘𝑦)) → ∃𝑥 ∈ ω 𝑦𝑥)
2520, 22, 24syl2anr 596 . . . . . . . . 9 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → ∃𝑥 ∈ ω 𝑦𝑥)
26 isfi 8388 . . . . . . . . 9 (𝑦 ∈ Fin ↔ ∃𝑥 ∈ ω 𝑦𝑥)
2725, 26sylibr 235 . . . . . . . 8 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
28 finnum 9230 . . . . . . . . 9 (𝑦 ∈ Fin → 𝑦 ∈ dom card)
29 ficardom 9243 . . . . . . . . 9 (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
3028, 29jca 512 . . . . . . . 8 (𝑦 ∈ Fin → (𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω))
3127, 30impbii 210 . . . . . . 7 ((𝑦 ∈ dom card ∧ (card‘𝑦) ∈ ω) ↔ 𝑦 ∈ Fin)
3219, 31bitri 276 . . . . . 6 (𝑦 ∈ (card “ ω) ↔ 𝑦 ∈ Fin)
3332eqriv 2794 . . . . 5 (card “ ω) = Fin
3412, 15, 333eqtri 2825 . . . 4 dom (𝐺 ∘ card) = Fin
35 df-fn 6235 . . . 4 ((𝐺 ∘ card) Fn Fin ↔ (Fun (𝐺 ∘ card) ∧ dom (𝐺 ∘ card) = Fin))
3611, 34, 35mpbir2an 707 . . 3 (𝐺 ∘ card) Fn Fin
37 hashkf.2 . . . 4 𝐾 = (𝐺 ∘ card)
3837fneq1i 6327 . . 3 (𝐾 Fn Fin ↔ (𝐺 ∘ card) Fn Fin)
3936, 38mpbir 232 . 2 𝐾 Fn Fin
4037fveq1i 6546 . . . . 5 (𝐾𝑦) = ((𝐺 ∘ card)‘𝑦)
41 fvco 6633 . . . . . 6 ((Fun card ∧ 𝑦 ∈ dom card) → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
429, 28, 41sylancr 587 . . . . 5 (𝑦 ∈ Fin → ((𝐺 ∘ card)‘𝑦) = (𝐺‘(card‘𝑦)))
4340, 42syl5eq 2845 . . . 4 (𝑦 ∈ Fin → (𝐾𝑦) = (𝐺‘(card‘𝑦)))
442hashgf1o 13193 . . . . . . 7 𝐺:ω–1-1-onto→ℕ0
45 f1of 6490 . . . . . . 7 (𝐺:ω–1-1-onto→ℕ0𝐺:ω⟶ℕ0)
4644, 45ax-mp 5 . . . . . 6 𝐺:ω⟶ℕ0
4746ffvelrni 6722 . . . . 5 ((card‘𝑦) ∈ ω → (𝐺‘(card‘𝑦)) ∈ ℕ0)
4829, 47syl 17 . . . 4 (𝑦 ∈ Fin → (𝐺‘(card‘𝑦)) ∈ ℕ0)
4943, 48eqeltrd 2885 . . 3 (𝑦 ∈ Fin → (𝐾𝑦) ∈ ℕ0)
5049rgen 3117 . 2 𝑦 ∈ Fin (𝐾𝑦) ∈ ℕ0
51 ffnfv 6752 . 2 (𝐾:Fin⟶ℕ0 ↔ (𝐾 Fn Fin ∧ ∀𝑦 ∈ Fin (𝐾𝑦) ∈ ℕ0))
5239, 50, 51mpbir2an 707 1 𝐾:Fin⟶ℕ0
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1525  wcel 2083  {cab 2777  wral 3107  wrex 3108  Vcvv 3440   class class class wbr 4968  cmpt 5047  ccnv 5449  dom cdm 5450  cres 5452  cima 5453  ccom 5454  Oncon0 6073  Fun wfun 6226   Fn wfn 6227  wf 6228  1-1-ontowf1o 6231  cfv 6232  (class class class)co 7023  ωcom 7443  reccrdg 7904  cen 8361  Fincfn 8364  cardccrd 9217  0cc0 10390  1c1 10391   + caddc 10393  0cn0 11751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-n0 11752  df-z 11836  df-uz 12098
This theorem is referenced by:  hashgval  13547  hashinf  13549  hashfxnn0  13551
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