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Theorem hashgval 12453
Description: The value of the  # function in terms of the mapping  G from  om to  NN0. The proof avoids the use of ax-ac 8870. (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 5107 . . . . . 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 2402 . . . . . . . . . 10  |-  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  =  ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om )
3 eqid 2402 . . . . . . . . . 10  |-  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )  =  ( ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  o.  card )
42, 3hashkf 12452 . . . . . . . . 9  |-  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card ) : Fin --> NN0
5 ffn 5713 . . . . . . . . 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 5670 . . . . . . . . 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 3631 . . . . . . . . . 10  |-  ( ( _V  \  Fin )  i^i  Fin )  =  ( Fin  i^i  ( _V 
\  Fin ) )
9 disjdif 3843 . . . . . . . . . 10  |-  ( Fin 
i^i  ( _V  \  Fin ) )  =  (/)
108, 9eqtri 2431 . . . . . . . . 9  |-  ( ( _V  \  Fin )  i^i  Fin )  =  (/)
11 pnfex 11374 . . . . . . . . . . 11  |- +oo  e.  _V
1211fconst 5753 . . . . . . . . . 10  |-  ( ( _V  \  Fin )  X.  { +oo } ) : ( _V  \  Fin ) --> { +oo }
13 ffn 5713 . . . . . . . . . 10  |-  ( ( ( _V  \  Fin )  X.  { +oo }
) : ( _V 
\  Fin ) --> { +oo }  ->  ( ( _V 
\  Fin )  X.  { +oo } )  Fn  ( _V  \  Fin ) )
14 fnresdisj 5671 . . . . . . . . . 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 208 . . . . . . . 8  |-  ( ( ( _V  \  Fin )  X.  { +oo }
)  |`  Fin )  =  (/)
177, 16uneq12i 3594 . . . . . . 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 3763 . . . . . . 7  |-  ( ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )  u.  (/) )  =  ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om )  o.  card )
1917, 18eqtri 2431 . . . . . 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 2431 . . . . 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 12451 . . . . . 6  |-  #  =  ( ( ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  o.  card )  u.  (
( _V  \  Fin )  X.  { +oo }
) )
2221reseq1i 5089 . . . . 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 4982 . . . . 5  |-  ( G  o.  card )  =  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )
2520, 22, 243eqtr4i 2441 . . . 4  |-  ( #  |` 
Fin )  =  ( G  o.  card )
2625fveq1i 5849 . . 3  |-  ( (
#  |`  Fin ) `  A )  =  ( ( G  o.  card ) `  A )
27 cardf2 8355 . . . . 5  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
28 ffun 5715 . . . . 5  |-  ( card
: { x  |  E. y  e.  On  y  ~~  x } --> On  ->  Fun 
card )
2927, 28ax-mp 5 . . . 4  |-  Fun  card
30 finnum 8360 . . . 4  |-  ( A  e.  Fin  ->  A  e.  dom  card )
31 fvco 5924 . . . 4  |-  ( ( Fun  card  /\  A  e. 
dom  card )  ->  (
( G  o.  card ) `  A )  =  ( G `  ( card `  A )
) )
3229, 30, 31sylancr 661 . . 3  |-  ( A  e.  Fin  ->  (
( G  o.  card ) `  A )  =  ( G `  ( card `  A )
) )
3326, 32syl5eq 2455 . 2  |-  ( A  e.  Fin  ->  (
( #  |`  Fin ) `  A )  =  ( G `  ( card `  A ) ) )
34 fvres 5862 . 2  |-  ( A  e.  Fin  ->  (
( #  |`  Fin ) `  A )  =  (
# `  A )
)
3533, 34eqtr3d 2445 1  |-  ( A  e.  Fin  ->  ( G `  ( card `  A ) )  =  ( # `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405    e. wcel 1842   {cab 2387   E.wrex 2754   _Vcvv 3058    \ cdif 3410    u. cun 3411    i^i cin 3412   (/)c0 3737   {csn 3971   class class class wbr 4394    |-> cmpt 4452    X. cxp 4820   dom cdm 4822    |` cres 4824    o. ccom 4826   Oncon0 5409   Fun wfun 5562    Fn wfn 5563   -->wf 5564   ` cfv 5568  (class class class)co 6277   omcom 6682   reccrdg 7111    ~~ cen 7550   Fincfn 7553   cardccrd 8347   0cc0 9521   1c1 9522    + caddc 9524   +oocpnf 9654   NN0cn0 10835   #chash 12450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-n0 10836  df-z 10905  df-uz 11127  df-hash 12451
This theorem is referenced by:  hashginv  12454  hashfz1  12464  hashen  12465  hashcard  12472  hashcl  12473  hashgval2  12492  hashdom  12493  hashun  12496  fz1isolem  12557
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