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Theorem hashkf 11548
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 6629 . . . . . . 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 5480 . . . . . . 7  |-  ( G  Fn  om  <->  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  Fn  om )
41, 3mpbir 201 . . . . . 6  |-  G  Fn  om
5 fnfun 5483 . . . . . 6  |-  ( G  Fn  om  ->  Fun  G )
64, 5ax-mp 8 . . . . 5  |-  Fun  G
7 cardf2 7764 . . . . . 6  |-  card : {
y  |  E. x  e.  On  x  ~~  y }
--> On
8 ffun 5534 . . . . . 6  |-  ( card
: { y  |  E. x  e.  On  x  ~~  y } --> On  ->  Fun 
card )
97, 8ax-mp 8 . . . . 5  |-  Fun  card
10 funco 5432 . . . . 5  |-  ( ( Fun  G  /\  Fun  card )  ->  Fun  ( G  o.  card ) )
116, 9, 10mp2an 654 . . . 4  |-  Fun  ( G  o.  card )
12 dmco 5319 . . . . 5  |-  dom  ( G  o.  card )  =  ( `' card " dom  G )
13 fndm 5485 . . . . . . 7  |-  ( G  Fn  om  ->  dom  G  =  om )
144, 13ax-mp 8 . . . . . 6  |-  dom  G  =  om
1514imaeq2i 5142 . . . . 5  |-  ( `'
card " dom  G )  =  ( `' card " om )
16 funfn 5423 . . . . . . . . 9  |-  ( Fun 
card 
<-> 
card  Fn  dom  card )
179, 16mpbi 200 . . . . . . . 8  |-  card  Fn  dom  card
18 elpreima 5790 . . . . . . . 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 20 . . . . . . . . . 10  |-  ( (
card `  y )  e.  om  ->  ( card `  y )  e.  om )
21 cardid2 7774 . . . . . . . . . . 11  |-  ( y  e.  dom  card  ->  (
card `  y )  ~~  y )
2221ensymd 7095 . . . . . . . . . 10  |-  ( y  e.  dom  card  ->  y 
~~  ( card `  y
) )
23 breq2 4158 . . . . . . . . . . 11  |-  ( x  =  ( card `  y
)  ->  ( y  ~~  x  <->  y  ~~  ( card `  y ) ) )
2423rspcev 2996 . . . . . . . . . 10  |-  ( ( ( card `  y
)  e.  om  /\  y  ~~  ( card `  y
) )  ->  E. x  e.  om  y  ~~  x
)
2520, 22, 24syl2anr 465 . . . . . . . . 9  |-  ( ( y  e.  dom  card  /\  ( card `  y
)  e.  om )  ->  E. x  e.  om  y  ~~  x )
26 isfi 7068 . . . . . . . . 9  |-  ( y  e.  Fin  <->  E. x  e.  om  y  ~~  x
)
2725, 26sylibr 204 . . . . . . . 8  |-  ( ( y  e.  dom  card  /\  ( card `  y
)  e.  om )  ->  y  e.  Fin )
28 finnum 7769 . . . . . . . . 9  |-  ( y  e.  Fin  ->  y  e.  dom  card )
29 ficardom 7782 . . . . . . . . 9  |-  ( y  e.  Fin  ->  ( card `  y )  e. 
om )
3028, 29jca 519 . . . . . . . 8  |-  ( y  e.  Fin  ->  (
y  e.  dom  card  /\  ( card `  y
)  e.  om )
)
3127, 30impbii 181 . . . . . . 7  |-  ( ( y  e.  dom  card  /\  ( card `  y
)  e.  om )  <->  y  e.  Fin )
3219, 31bitri 241 . . . . . 6  |-  ( y  e.  ( `' card " om )  <->  y  e.  Fin )
3332eqriv 2385 . . . . 5  |-  ( `'
card " om )  =  Fin
3412, 15, 333eqtri 2412 . . . 4  |-  dom  ( G  o.  card )  =  Fin
35 df-fn 5398 . . . 4  |-  ( ( G  o.  card )  Fn  Fin  <->  ( Fun  ( G  o.  card )  /\  dom  ( G  o.  card )  =  Fin )
)
3611, 34, 35mpbir2an 887 . . 3  |-  ( G  o.  card )  Fn  Fin
37 hashkf.2 . . . 4  |-  K  =  ( G  o.  card )
3837fneq1i 5480 . . 3  |-  ( K  Fn  Fin  <->  ( G  o.  card )  Fn  Fin )
3936, 38mpbir 201 . 2  |-  K  Fn  Fin
4037fveq1i 5670 . . . . 5  |-  ( K `
 y )  =  ( ( G  o.  card ) `  y )
41 fvco 5739 . . . . . 6  |-  ( ( Fun  card  /\  y  e.  dom  card )  ->  (
( G  o.  card ) `  y )  =  ( G `  ( card `  y )
) )
429, 28, 41sylancr 645 . . . . 5  |-  ( y  e.  Fin  ->  (
( G  o.  card ) `  y )  =  ( G `  ( card `  y )
) )
4340, 42syl5eq 2432 . . . 4  |-  ( y  e.  Fin  ->  ( K `  y )  =  ( G `  ( card `  y )
) )
442hashgf1o 11238 . . . . . . 7  |-  G : om
-1-1-onto-> NN0
45 f1of 5615 . . . . . . 7  |-  ( G : om -1-1-onto-> NN0  ->  G : om
--> NN0 )
4644, 45ax-mp 8 . . . . . 6  |-  G : om
--> NN0
4746ffvelrni 5809 . . . . 5  |-  ( (
card `  y )  e.  om  ->  ( G `  ( card `  y
) )  e.  NN0 )
4829, 47syl 16 . . . 4  |-  ( y  e.  Fin  ->  ( G `  ( card `  y ) )  e. 
NN0 )
4943, 48eqeltrd 2462 . . 3  |-  ( y  e.  Fin  ->  ( K `  y )  e.  NN0 )
5049rgen 2715 . 2  |-  A. y  e.  Fin  ( K `  y )  e.  NN0
51 ffnfv 5834 . 2  |-  ( K : Fin --> NN0  <->  ( K  Fn  Fin  /\  A. y  e.  Fin  ( K `  y )  e.  NN0 ) )
5239, 50, 51mpbir2an 887 1  |-  K : Fin
--> NN0
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2374   A.wral 2650   E.wrex 2651   _Vcvv 2900   class class class wbr 4154    e. cmpt 4208   Oncon0 4523   omcom 4786   `'ccnv 4818   dom cdm 4819    |` cres 4821   "cima 4822    o. ccom 4823   Fun wfun 5389    Fn wfn 5390   -->wf 5391   -1-1-onto->wf1o 5394   ` cfv 5395  (class class class)co 6021   reccrdg 6604    ~~ cen 7043   Fincfn 7046   cardccrd 7756   0cc0 8924   1c1 8925    + caddc 8927   NN0cn0 10154
This theorem is referenced by:  hashgval  11549  hashinf  11551  hashf  11553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422
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