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Theorem hashkf 12364
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 7092 . . . . . . 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 5668 . . . . . . 7  |-  ( G  Fn  om  <->  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  Fn  om )
41, 3mpbir 209 . . . . . 6  |-  G  Fn  om
5 fnfun 5671 . . . . . 6  |-  ( G  Fn  om  ->  Fun  G )
64, 5ax-mp 5 . . . . 5  |-  Fun  G
7 cardf2 8315 . . . . . 6  |-  card : {
y  |  E. x  e.  On  x  ~~  y }
--> On
8 ffun 5726 . . . . . 6  |-  ( card
: { y  |  E. x  e.  On  x  ~~  y } --> On  ->  Fun 
card )
97, 8ax-mp 5 . . . . 5  |-  Fun  card
10 funco 5619 . . . . 5  |-  ( ( Fun  G  /\  Fun  card )  ->  Fun  ( G  o.  card ) )
116, 9, 10mp2an 672 . . . 4  |-  Fun  ( G  o.  card )
12 dmco 5508 . . . . 5  |-  dom  ( G  o.  card )  =  ( `' card " dom  G )
13 fndm 5673 . . . . . . 7  |-  ( G  Fn  om  ->  dom  G  =  om )
144, 13ax-mp 5 . . . . . 6  |-  dom  G  =  om
1514imaeq2i 5328 . . . . 5  |-  ( `'
card " dom  G )  =  ( `' card " om )
16 funfn 5610 . . . . . . . . 9  |-  ( Fun 
card 
<-> 
card  Fn  dom  card )
179, 16mpbi 208 . . . . . . . 8  |-  card  Fn  dom  card
18 elpreima 5994 . . . . . . . 8  |-  ( card 
Fn  dom  card  ->  (
y  e.  ( `'
card " om )  <->  ( y  e.  dom  card  /\  ( card `  y )  e. 
om ) ) )
1917, 18ax-mp 5 . . . . . . 7  |-  ( y  e.  ( `' card " om )  <->  ( y  e.  dom  card  /\  ( card `  y )  e. 
om ) )
20 id 22 . . . . . . . . . 10  |-  ( (
card `  y )  e.  om  ->  ( card `  y )  e.  om )
21 cardid2 8325 . . . . . . . . . . 11  |-  ( y  e.  dom  card  ->  (
card `  y )  ~~  y )
2221ensymd 7558 . . . . . . . . . 10  |-  ( y  e.  dom  card  ->  y 
~~  ( card `  y
) )
23 breq2 4446 . . . . . . . . . . 11  |-  ( x  =  ( card `  y
)  ->  ( y  ~~  x  <->  y  ~~  ( card `  y ) ) )
2423rspcev 3209 . . . . . . . . . 10  |-  ( ( ( card `  y
)  e.  om  /\  y  ~~  ( card `  y
) )  ->  E. x  e.  om  y  ~~  x
)
2520, 22, 24syl2anr 478 . . . . . . . . 9  |-  ( ( y  e.  dom  card  /\  ( card `  y
)  e.  om )  ->  E. x  e.  om  y  ~~  x )
26 isfi 7531 . . . . . . . . 9  |-  ( y  e.  Fin  <->  E. x  e.  om  y  ~~  x
)
2725, 26sylibr 212 . . . . . . . 8  |-  ( ( y  e.  dom  card  /\  ( card `  y
)  e.  om )  ->  y  e.  Fin )
28 finnum 8320 . . . . . . . . 9  |-  ( y  e.  Fin  ->  y  e.  dom  card )
29 ficardom 8333 . . . . . . . . 9  |-  ( y  e.  Fin  ->  ( card `  y )  e. 
om )
3028, 29jca 532 . . . . . . . 8  |-  ( y  e.  Fin  ->  (
y  e.  dom  card  /\  ( card `  y
)  e.  om )
)
3127, 30impbii 188 . . . . . . 7  |-  ( ( y  e.  dom  card  /\  ( card `  y
)  e.  om )  <->  y  e.  Fin )
3219, 31bitri 249 . . . . . 6  |-  ( y  e.  ( `' card " om )  <->  y  e.  Fin )
3332eqriv 2458 . . . . 5  |-  ( `'
card " om )  =  Fin
3412, 15, 333eqtri 2495 . . . 4  |-  dom  ( G  o.  card )  =  Fin
35 df-fn 5584 . . . 4  |-  ( ( G  o.  card )  Fn  Fin  <->  ( Fun  ( G  o.  card )  /\  dom  ( G  o.  card )  =  Fin )
)
3611, 34, 35mpbir2an 913 . . 3  |-  ( G  o.  card )  Fn  Fin
37 hashkf.2 . . . 4  |-  K  =  ( G  o.  card )
3837fneq1i 5668 . . 3  |-  ( K  Fn  Fin  <->  ( G  o.  card )  Fn  Fin )
3936, 38mpbir 209 . 2  |-  K  Fn  Fin
4037fveq1i 5860 . . . . 5  |-  ( K `
 y )  =  ( ( G  o.  card ) `  y )
41 fvco 5936 . . . . . 6  |-  ( ( Fun  card  /\  y  e.  dom  card )  ->  (
( G  o.  card ) `  y )  =  ( G `  ( card `  y )
) )
429, 28, 41sylancr 663 . . . . 5  |-  ( y  e.  Fin  ->  (
( G  o.  card ) `  y )  =  ( G `  ( card `  y )
) )
4340, 42syl5eq 2515 . . . 4  |-  ( y  e.  Fin  ->  ( K `  y )  =  ( G `  ( card `  y )
) )
442hashgf1o 12039 . . . . . . 7  |-  G : om
-1-1-onto-> NN0
45 f1of 5809 . . . . . . 7  |-  ( G : om -1-1-onto-> NN0  ->  G : om
--> NN0 )
4644, 45ax-mp 5 . . . . . 6  |-  G : om
--> NN0
4746ffvelrni 6013 . . . . 5  |-  ( (
card `  y )  e.  om  ->  ( G `  ( card `  y
) )  e.  NN0 )
4829, 47syl 16 . . . 4  |-  ( y  e.  Fin  ->  ( G `  ( card `  y ) )  e. 
NN0 )
4943, 48eqeltrd 2550 . . 3  |-  ( y  e.  Fin  ->  ( K `  y )  e.  NN0 )
5049rgen 2819 . 2  |-  A. y  e.  Fin  ( K `  y )  e.  NN0
51 ffnfv 6040 . 2  |-  ( K : Fin --> NN0  <->  ( K  Fn  Fin  /\  A. y  e.  Fin  ( K `  y )  e.  NN0 ) )
5239, 50, 51mpbir2an 913 1  |-  K : Fin
--> NN0
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   {cab 2447   A.wral 2809   E.wrex 2810   _Vcvv 3108   class class class wbr 4442    |-> cmpt 4500   Oncon0 4873   `'ccnv 4993   dom cdm 4994    |` cres 4996   "cima 4997    o. ccom 4998   Fun wfun 5575    Fn wfn 5576   -->wf 5577   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277   omcom 6673   reccrdg 7067    ~~ cen 7505   Fincfn 7508   cardccrd 8307   0cc0 9483   1c1 9484    + caddc 9486   NN0cn0 10786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074
This theorem is referenced by:  hashgval  12365  hashinf  12367  hashf  12369
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