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Theorem hashkf 11355
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 6463 . . . . . . 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 5354 . . . . . . 7  |-  ( G  Fn  om  <->  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  Fn  om )
41, 3mpbir 200 . . . . . 6  |-  G  Fn  om
5 fnfun 5357 . . . . . 6  |-  ( G  Fn  om  ->  Fun  G )
64, 5ax-mp 8 . . . . 5  |-  Fun  G
7 cardf2 7592 . . . . . 6  |-  card : {
y  |  E. x  e.  On  x  ~~  y }
--> On
8 ffun 5407 . . . . . 6  |-  ( card
: { y  |  E. x  e.  On  x  ~~  y } --> On  ->  Fun 
card )
97, 8ax-mp 8 . . . . 5  |-  Fun  card
10 funco 5308 . . . . 5  |-  ( ( Fun  G  /\  Fun  card )  ->  Fun  ( G  o.  card ) )
116, 9, 10mp2an 653 . . . 4  |-  Fun  ( G  o.  card )
12 dmco 5197 . . . . 5  |-  dom  ( G  o.  card )  =  ( `' card " dom  G )
13 fndm 5359 . . . . . . 7  |-  ( G  Fn  om  ->  dom  G  =  om )
144, 13ax-mp 8 . . . . . 6  |-  dom  G  =  om
1514imaeq2i 5026 . . . . 5  |-  ( `'
card " dom  G )  =  ( `' card " om )
16 funfn 5299 . . . . . . . . 9  |-  ( Fun 
card 
<-> 
card  Fn  dom  card )
179, 16mpbi 199 . . . . . . . 8  |-  card  Fn  dom  card
18 elpreima 5661 . . . . . . . 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 7602 . . . . . . . . . . 11  |-  ( y  e.  dom  card  ->  (
card `  y )  ~~  y )
22 ensym 6926 . . . . . . . . . . 11  |-  ( (
card `  y )  ~~  y  ->  y  ~~  ( card `  y )
)
2321, 22syl 15 . . . . . . . . . 10  |-  ( y  e.  dom  card  ->  y 
~~  ( card `  y
) )
24 breq2 4043 . . . . . . . . . . 11  |-  ( x  =  ( card `  y
)  ->  ( y  ~~  x  <->  y  ~~  ( card `  y ) ) )
2524rspcev 2897 . . . . . . . . . 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 6901 . . . . . . . . 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 7597 . . . . . . . . 9  |-  ( y  e.  Fin  ->  y  e.  dom  card )
30 ficardom 7610 . . . . . . . . 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 2293 . . . . 5  |-  ( `'
card " om )  =  Fin
3512, 15, 343eqtri 2320 . . . 4  |-  dom  ( G  o.  card )  =  Fin
36 df-fn 5274 . . . 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 5354 . . 3  |-  ( K  Fn  Fin  <->  ( G  o.  card )  Fn  Fin )
4037, 39mpbir 200 . 2  |-  K  Fn  Fin
4138fveq1i 5542 . . . . 5  |-  ( K `
 y )  =  ( ( G  o.  card ) `  y )
42 fvco 5611 . . . . . 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 2340 . . . 4  |-  ( y  e.  Fin  ->  ( K `  y )  =  ( G `  ( card `  y )
) )
452hashgf1o 11049 . . . . . . 7  |-  G : om
-1-1-onto-> NN0
46 f1of 5488 . . . . . . 7  |-  ( G : om -1-1-onto-> NN0  ->  G : om
--> NN0 )
4745, 46ax-mp 8 . . . . . 6  |-  G : om
--> NN0
4847ffvelrni 5680 . . . . 5  |-  ( (
card `  y )  e.  om  ->  ( G `  ( card `  y
) )  e.  NN0 )
4930, 48syl 15 . . . 4  |-  ( y  e.  Fin  ->  ( G `  ( card `  y ) )  e. 
NN0 )
5044, 49eqeltrd 2370 . . 3  |-  ( y  e.  Fin  ->  ( K `  y )  e.  NN0 )
5150rgen 2621 . 2  |-  A. y  e.  Fin  ( K `  y )  e.  NN0
52 ffnfv 5701 . 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 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   Oncon0 4408   omcom 4672   `'ccnv 4704   dom cdm 4705    |` cres 4707   "cima 4708    o. ccom 4709   Fun wfun 5265    Fn wfn 5266   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   reccrdg 6438    ~~ cen 6876   Fincfn 6879   cardccrd 7584   0cc0 8753   1c1 8754    + caddc 8756   NN0cn0 9981
This theorem is referenced by:  hashgval  11356  hashinf  11358  hashf  11360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247
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