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Theorem hashkf 11612
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 6684 . . . . . . 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 5531 . . . . . . 7  |-  ( G  Fn  om  <->  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  Fn  om )
41, 3mpbir 201 . . . . . 6  |-  G  Fn  om
5 fnfun 5534 . . . . . 6  |-  ( G  Fn  om  ->  Fun  G )
64, 5ax-mp 8 . . . . 5  |-  Fun  G
7 cardf2 7822 . . . . . 6  |-  card : {
y  |  E. x  e.  On  x  ~~  y }
--> On
8 ffun 5585 . . . . . 6  |-  ( card
: { y  |  E. x  e.  On  x  ~~  y } --> On  ->  Fun 
card )
97, 8ax-mp 8 . . . . 5  |-  Fun  card
10 funco 5483 . . . . 5  |-  ( ( Fun  G  /\  Fun  card )  ->  Fun  ( G  o.  card ) )
116, 9, 10mp2an 654 . . . 4  |-  Fun  ( G  o.  card )
12 dmco 5370 . . . . 5  |-  dom  ( G  o.  card )  =  ( `' card " dom  G )
13 fndm 5536 . . . . . . 7  |-  ( G  Fn  om  ->  dom  G  =  om )
144, 13ax-mp 8 . . . . . 6  |-  dom  G  =  om
1514imaeq2i 5193 . . . . 5  |-  ( `'
card " dom  G )  =  ( `' card " om )
16 funfn 5474 . . . . . . . . 9  |-  ( Fun 
card 
<-> 
card  Fn  dom  card )
179, 16mpbi 200 . . . . . . . 8  |-  card  Fn  dom  card
18 elpreima 5842 . . . . . . . 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 7832 . . . . . . . . . . 11  |-  ( y  e.  dom  card  ->  (
card `  y )  ~~  y )
2221ensymd 7150 . . . . . . . . . 10  |-  ( y  e.  dom  card  ->  y 
~~  ( card `  y
) )
23 breq2 4208 . . . . . . . . . . 11  |-  ( x  =  ( card `  y
)  ->  ( y  ~~  x  <->  y  ~~  ( card `  y ) ) )
2423rspcev 3044 . . . . . . . . . 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 7123 . . . . . . . . 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 7827 . . . . . . . . 9  |-  ( y  e.  Fin  ->  y  e.  dom  card )
29 ficardom 7840 . . . . . . . . 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 2432 . . . . 5  |-  ( `'
card " om )  =  Fin
3412, 15, 333eqtri 2459 . . . 4  |-  dom  ( G  o.  card )  =  Fin
35 df-fn 5449 . . . 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 5531 . . 3  |-  ( K  Fn  Fin  <->  ( G  o.  card )  Fn  Fin )
3936, 38mpbir 201 . 2  |-  K  Fn  Fin
4037fveq1i 5721 . . . . 5  |-  ( K `
 y )  =  ( ( G  o.  card ) `  y )
41 fvco 5791 . . . . . 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 2479 . . . 4  |-  ( y  e.  Fin  ->  ( K `  y )  =  ( G `  ( card `  y )
) )
442hashgf1o 11302 . . . . . . 7  |-  G : om
-1-1-onto-> NN0
45 f1of 5666 . . . . . . 7  |-  ( G : om -1-1-onto-> NN0  ->  G : om
--> NN0 )
4644, 45ax-mp 8 . . . . . 6  |-  G : om
--> NN0
4746ffvelrni 5861 . . . . 5  |-  ( (
card `  y )  e.  om  ->  ( G `  ( card `  y
) )  e.  NN0 )
4829, 47syl 16 . . . 4  |-  ( y  e.  Fin  ->  ( G `  ( card `  y ) )  e. 
NN0 )
4943, 48eqeltrd 2509 . . 3  |-  ( y  e.  Fin  ->  ( K `  y )  e.  NN0 )
5049rgen 2763 . 2  |-  A. y  e.  Fin  ( K `  y )  e.  NN0
51 ffnfv 5886 . 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 1652    e. wcel 1725   {cab 2421   A.wral 2697   E.wrex 2698   _Vcvv 2948   class class class wbr 4204    e. cmpt 4258   Oncon0 4573   omcom 4837   `'ccnv 4869   dom cdm 4870    |` cres 4872   "cima 4873    o. ccom 4874   Fun wfun 5440    Fn wfn 5441   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   reccrdg 6659    ~~ cen 7098   Fincfn 7101   cardccrd 7814   0cc0 8982   1c1 8983    + caddc 8985   NN0cn0 10213
This theorem is referenced by:  hashgval  11613  hashinf  11615  hashf  11617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481
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