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Theorem hashgval 12556
Description: The value of the  # function in terms of the mapping  G from  om to  NN0. The proof avoids the use of ax-ac 8907. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 26-Dec-2014.)
Hypothesis
Ref Expression
hashgval.1  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
Assertion
Ref Expression
hashgval  |-  ( A  e.  Fin  ->  ( G `  ( card `  A ) )  =  ( # `  A
) )
Distinct variable group:    x, A
Allowed substitution hint:    G( x)

Proof of Theorem hashgval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 resundir 5125 . . . . . 6  |-  ( ( ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om )  o.  card )  u.  (
( _V  \  Fin )  X.  { +oo }
) )  |`  Fin )  =  ( ( ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )  |`  Fin )  u.  ( ( ( _V 
\  Fin )  X.  { +oo } )  |`  Fin )
)
2 eqid 2471 . . . . . . . . . 10  |-  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  =  ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om )
3 eqid 2471 . . . . . . . . . 10  |-  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )  =  ( ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  o.  card )
42, 3hashkf 12555 . . . . . . . . 9  |-  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card ) : Fin --> NN0
5 ffn 5739 . . . . . . . . 9  |-  ( ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card ) : Fin --> NN0  ->  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )  Fn  Fin )
6 fnresdm 5695 . . . . . . . . 9  |-  ( ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )  Fn  Fin  ->  (
( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om )  o.  card )  |`  Fin )  =  ( ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  o.  card ) )
74, 5, 6mp2b 10 . . . . . . . 8  |-  ( ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )  |`  Fin )  =  ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om )  o.  card )
8 incom 3616 . . . . . . . . . 10  |-  ( ( _V  \  Fin )  i^i  Fin )  =  ( Fin  i^i  ( _V 
\  Fin ) )
9 disjdif 3830 . . . . . . . . . 10  |-  ( Fin 
i^i  ( _V  \  Fin ) )  =  (/)
108, 9eqtri 2493 . . . . . . . . 9  |-  ( ( _V  \  Fin )  i^i  Fin )  =  (/)
11 pnfex 11436 . . . . . . . . . . 11  |- +oo  e.  _V
1211fconst 5782 . . . . . . . . . 10  |-  ( ( _V  \  Fin )  X.  { +oo } ) : ( _V  \  Fin ) --> { +oo }
13 ffn 5739 . . . . . . . . . 10  |-  ( ( ( _V  \  Fin )  X.  { +oo }
) : ( _V 
\  Fin ) --> { +oo }  ->  ( ( _V 
\  Fin )  X.  { +oo } )  Fn  ( _V  \  Fin ) )
14 fnresdisj 5696 . . . . . . . . . 10  |-  ( ( ( _V  \  Fin )  X.  { +oo }
)  Fn  ( _V 
\  Fin )  ->  (
( ( _V  \  Fin )  i^i  Fin )  =  (/)  <->  ( ( ( _V  \  Fin )  X.  { +oo } )  |`  Fin )  =  (/) ) )
1512, 13, 14mp2b 10 . . . . . . . . 9  |-  ( ( ( _V  \  Fin )  i^i  Fin )  =  (/) 
<->  ( ( ( _V 
\  Fin )  X.  { +oo } )  |`  Fin )  =  (/) )
1610, 15mpbi 213 . . . . . . . 8  |-  ( ( ( _V  \  Fin )  X.  { +oo }
)  |`  Fin )  =  (/)
177, 16uneq12i 3577 . . . . . . 7  |-  ( ( ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om )  o.  card )  |`  Fin )  u.  ( ( ( _V 
\  Fin )  X.  { +oo } )  |`  Fin )
)  =  ( ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )  u.  (/) )
18 un0 3762 . . . . . . 7  |-  ( ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )  u.  (/) )  =  ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om )  o.  card )
1917, 18eqtri 2493 . . . . . 6  |-  ( ( ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om )  o.  card )  |`  Fin )  u.  ( ( ( _V 
\  Fin )  X.  { +oo } )  |`  Fin )
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )
201, 19eqtri 2493 . . . . 5  |-  ( ( ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om )  o.  card )  u.  (
( _V  \  Fin )  X.  { +oo }
) )  |`  Fin )  =  ( ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  o.  card )
21 df-hash 12554 . . . . . 6  |-  #  =  ( ( ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  o.  card )  u.  (
( _V  \  Fin )  X.  { +oo }
) )
2221reseq1i 5107 . . . . 5  |-  ( #  |` 
Fin )  =  ( ( ( ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  o.  card )  u.  (
( _V  \  Fin )  X.  { +oo }
) )  |`  Fin )
23 hashgval.1 . . . . . 6  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
2423coeq1i 4999 . . . . 5  |-  ( G  o.  card )  =  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )
2520, 22, 243eqtr4i 2503 . . . 4  |-  ( #  |` 
Fin )  =  ( G  o.  card )
2625fveq1i 5880 . . 3  |-  ( (
#  |`  Fin ) `  A )  =  ( ( G  o.  card ) `  A )
27 cardf2 8395 . . . . 5  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
28 ffun 5742 . . . . 5  |-  ( card
: { x  |  E. y  e.  On  y  ~~  x } --> On  ->  Fun 
card )
2927, 28ax-mp 5 . . . 4  |-  Fun  card
30 finnum 8400 . . . 4  |-  ( A  e.  Fin  ->  A  e.  dom  card )
31 fvco 5956 . . . 4  |-  ( ( Fun  card  /\  A  e. 
dom  card )  ->  (
( G  o.  card ) `  A )  =  ( G `  ( card `  A )
) )
3229, 30, 31sylancr 676 . . 3  |-  ( A  e.  Fin  ->  (
( G  o.  card ) `  A )  =  ( G `  ( card `  A )
) )
3326, 32syl5eq 2517 . 2  |-  ( A  e.  Fin  ->  (
( #  |`  Fin ) `  A )  =  ( G `  ( card `  A ) ) )
34 fvres 5893 . 2  |-  ( A  e.  Fin  ->  (
( #  |`  Fin ) `  A )  =  (
# `  A )
)
3533, 34eqtr3d 2507 1  |-  ( A  e.  Fin  ->  ( G `  ( card `  A ) )  =  ( # `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904   {cab 2457   E.wrex 2757   _Vcvv 3031    \ cdif 3387    u. cun 3388    i^i cin 3389   (/)c0 3722   {csn 3959   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   dom cdm 4839    |` cres 4841    o. ccom 4843   Oncon0 5430   Fun wfun 5583    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   omcom 6711   reccrdg 7145    ~~ cen 7584   Fincfn 7587   cardccrd 8387   0cc0 9557   1c1 9558    + caddc 9560   +oocpnf 9690   NN0cn0 10893   #chash 12553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-hash 12554
This theorem is referenced by:  hashginv  12557  hashfz1  12567  hashen  12568  hashcard  12575  hashcl  12576  hashgval2  12595  hashdom  12596  hashun  12599  fz1isolem  12665
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