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Theorem hashkf 11339
Description: The finite part of the size function maps all finite sets to their cardinality, as members of  NN0. (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
Hypotheses
Ref Expression
hashgval.1  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
hashkf.2  |-  K  =  ( G  o.  card )
Assertion
Ref Expression
hashkf  |-  K : Fin
--> NN0

Proof of Theorem hashkf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 frfnom 6447 . . . . . . 7  |-  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  Fn  om
2 hashgval.1 . . . . . . . 8  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
32fneq1i 5338 . . . . . . 7  |-  ( G  Fn  om  <->  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  Fn  om )
41, 3mpbir 200 . . . . . 6  |-  G  Fn  om
5 fnfun 5341 . . . . . 6  |-  ( G  Fn  om  ->  Fun  G )
64, 5ax-mp 8 . . . . 5  |-  Fun  G
7 cardf2 7576 . . . . . 6  |-  card : {
y  |  E. x  e.  On  x  ~~  y }
--> On
8 ffun 5391 . . . . . 6  |-  ( card
: { y  |  E. x  e.  On  x  ~~  y } --> On  ->  Fun 
card )
97, 8ax-mp 8 . . . . 5  |-  Fun  card
10 funco 5292 . . . . 5  |-  ( ( Fun  G  /\  Fun  card )  ->  Fun  ( G  o.  card ) )
116, 9, 10mp2an 653 . . . 4  |-  Fun  ( G  o.  card )
12 dmco 5181 . . . . 5  |-  dom  ( G  o.  card )  =  ( `' card " dom  G )
13 fndm 5343 . . . . . . 7  |-  ( G  Fn  om  ->  dom  G  =  om )
144, 13ax-mp 8 . . . . . 6  |-  dom  G  =  om
1514imaeq2i 5010 . . . . 5  |-  ( `'
card " dom  G )  =  ( `' card " om )
16 funfn 5283 . . . . . . . . 9  |-  ( Fun 
card 
<-> 
card  Fn  dom  card )
179, 16mpbi 199 . . . . . . . 8  |-  card  Fn  dom  card
18 elpreima 5645 . . . . . . . 8  |-  ( card 
Fn  dom  card  ->  (
y  e.  ( `'
card " om )  <->  ( y  e.  dom  card  /\  ( card `  y )  e. 
om ) ) )
1917, 18ax-mp 8 . . . . . . 7  |-  ( y  e.  ( `' card " om )  <->  ( y  e.  dom  card  /\  ( card `  y )  e. 
om ) )
20 id 19 . . . . . . . . . 10  |-  ( (
card `  y )  e.  om  ->  ( card `  y )  e.  om )
21 cardid2 7586 . . . . . . . . . . 11  |-  ( y  e.  dom  card  ->  (
card `  y )  ~~  y )
22 ensym 6910 . . . . . . . . . . 11  |-  ( (
card `  y )  ~~  y  ->  y  ~~  ( card `  y )
)
2321, 22syl 15 . . . . . . . . . 10  |-  ( y  e.  dom  card  ->  y 
~~  ( card `  y
) )
24 breq2 4027 . . . . . . . . . . 11  |-  ( x  =  ( card `  y
)  ->  ( y  ~~  x  <->  y  ~~  ( card `  y ) ) )
2524rspcev 2884 . . . . . . . . . 10  |-  ( ( ( card `  y
)  e.  om  /\  y  ~~  ( card `  y
) )  ->  E. x  e.  om  y  ~~  x
)
2620, 23, 25syl2anr 464 . . . . . . . . 9  |-  ( ( y  e.  dom  card  /\  ( card `  y
)  e.  om )  ->  E. x  e.  om  y  ~~  x )
27 isfi 6885 . . . . . . . . 9  |-  ( y  e.  Fin  <->  E. x  e.  om  y  ~~  x
)
2826, 27sylibr 203 . . . . . . . 8  |-  ( ( y  e.  dom  card  /\  ( card `  y
)  e.  om )  ->  y  e.  Fin )
29 finnum 7581 . . . . . . . . 9  |-  ( y  e.  Fin  ->  y  e.  dom  card )
30 ficardom 7594 . . . . . . . . 9  |-  ( y  e.  Fin  ->  ( card `  y )  e. 
om )
3129, 30jca 518 . . . . . . . 8  |-  ( y  e.  Fin  ->  (
y  e.  dom  card  /\  ( card `  y
)  e.  om )
)
3228, 31impbii 180 . . . . . . 7  |-  ( ( y  e.  dom  card  /\  ( card `  y
)  e.  om )  <->  y  e.  Fin )
3319, 32bitri 240 . . . . . 6  |-  ( y  e.  ( `' card " om )  <->  y  e.  Fin )
3433eqriv 2280 . . . . 5  |-  ( `'
card " om )  =  Fin
3512, 15, 343eqtri 2307 . . . 4  |-  dom  ( G  o.  card )  =  Fin
36 df-fn 5258 . . . 4  |-  ( ( G  o.  card )  Fn  Fin  <->  ( Fun  ( G  o.  card )  /\  dom  ( G  o.  card )  =  Fin )
)
3711, 35, 36mpbir2an 886 . . 3  |-  ( G  o.  card )  Fn  Fin
38 hashkf.2 . . . 4  |-  K  =  ( G  o.  card )
3938fneq1i 5338 . . 3  |-  ( K  Fn  Fin  <->  ( G  o.  card )  Fn  Fin )
4037, 39mpbir 200 . 2  |-  K  Fn  Fin
4138fveq1i 5526 . . . . 5  |-  ( K `
 y )  =  ( ( G  o.  card ) `  y )
42 fvco 5595 . . . . . 6  |-  ( ( Fun  card  /\  y  e.  dom  card )  ->  (
( G  o.  card ) `  y )  =  ( G `  ( card `  y )
) )
439, 29, 42sylancr 644 . . . . 5  |-  ( y  e.  Fin  ->  (
( G  o.  card ) `  y )  =  ( G `  ( card `  y )
) )
4441, 43syl5eq 2327 . . . 4  |-  ( y  e.  Fin  ->  ( K `  y )  =  ( G `  ( card `  y )
) )
452hashgf1o 11033 . . . . . . 7  |-  G : om
-1-1-onto-> NN0
46 f1of 5472 . . . . . . 7  |-  ( G : om -1-1-onto-> NN0  ->  G : om
--> NN0 )
4745, 46ax-mp 8 . . . . . 6  |-  G : om
--> NN0
4847ffvelrni 5664 . . . . 5  |-  ( (
card `  y )  e.  om  ->  ( G `  ( card `  y
) )  e.  NN0 )
4930, 48syl 15 . . . 4  |-  ( y  e.  Fin  ->  ( G `  ( card `  y ) )  e. 
NN0 )
5044, 49eqeltrd 2357 . . 3  |-  ( y  e.  Fin  ->  ( K `  y )  e.  NN0 )
5150rgen 2608 . 2  |-  A. y  e.  Fin  ( K `  y )  e.  NN0
52 ffnfv 5685 . 2  |-  ( K : Fin --> NN0  <->  ( K  Fn  Fin  /\  A. y  e.  Fin  ( K `  y )  e.  NN0 ) )
5340, 51, 52mpbir2an 886 1  |-  K : Fin
--> NN0
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   Oncon0 4392   omcom 4656   `'ccnv 4688   dom cdm 4689    |` cres 4691   "cima 4692    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   reccrdg 6422    ~~ cen 6860   Fincfn 6863   cardccrd 7568   0cc0 8737   1c1 8738    + caddc 8740   NN0cn0 9965
This theorem is referenced by:  hashgval  11340  hashinf  11342  hashf  11344
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231
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