Theorem hashgval 13120

Description: The value of the # function in terms of the mapping G from om to NN0. The proof avoids the use of ax-ac 9281. (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.

Step	Hyp	Ref	Expression
1		resundir 5411	. . . . . 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 2622	. . . . . . . . . 10 |- ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om ) = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om )
3		eqid 2622	. . . . . . . . . 10 |- ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om ) o. card ) = ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om ) o. card )
4	2, 3	hashkf 13119	. . . . . . . . 9 |- ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om ) o. card ) : Fin --> NN0
5		ffn 6045	. . . . . . . . 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 6000	. . . . . . . . 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 ) )
7	4, 5, 6	mp2b 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 3805	. . . . . . . . . 10 |- ( ( _V \ Fin ) i^i Fin ) = ( Fin i^i ( _V \ Fin ) )
9		disjdif 4040	. . . . . . . . . 10 |- ( Fin i^i ( _V \ Fin ) ) = (/)
10	8, 9	eqtri 2644	. . . . . . . . 9 |- ( ( _V \ Fin ) i^i Fin ) = (/)
11		pnfex 10093	. . . . . . . . . . 11 |- +oo e. _V
12	11	fconst 6091	. . . . . . . . . 10 |- ( ( _V \ Fin ) X. { +oo } ) : ( _V \ Fin ) --> { +oo }
13		ffn 6045	. . . . . . . . . 10 |- ( ( ( _V \ Fin ) X. { +oo } ) : ( _V \ Fin ) --> { +oo } -> ( ( _V \ Fin ) X. { +oo } ) Fn ( _V \ Fin ) )
14		fnresdisj 6001	. . . . . . . . . 10 |- ( ( ( _V \ Fin ) X. { +oo } ) Fn ( _V \ Fin ) -> ( ( ( _V \ Fin ) i^i Fin ) = (/) <-> ( ( ( _V \ Fin ) X. { +oo } ) |` Fin ) = (/) ) )
15	12, 13, 14	mp2b 10	. . . . . . . . 9 |- ( ( ( _V \ Fin ) i^i Fin ) = (/) <-> ( ( ( _V \ Fin ) X. { +oo } ) |` Fin ) = (/) )
16	10, 15	mpbi 220	. . . . . . . 8 |- ( ( ( _V \ Fin ) X. { +oo } ) |` Fin ) = (/)
17	7, 16	uneq12i 3765	. . . . . . 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 3967	. . . . . . 7 |- ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om ) o. card ) u. (/) ) = ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om ) o. card )
19	17, 18	eqtri 2644	. . . . . 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 )
20	1, 19	eqtri 2644	. . . . 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 13118	. . . . . 6 |- # = ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om ) o. card ) u. ( ( _V \ Fin ) X. { +oo } ) )
22	21	reseq1i 5392	. . . . 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 )
24	23	coeq1i 5281	. . . . 5 |- ( G o. card ) = ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om ) o. card )
25	20, 22, 24	3eqtr4i 2654	. . . 4 |- ( # |` Fin ) = ( G o. card )
26	25	fveq1i 6192	. . 3 |- ( ( # |` Fin ) ` A ) = ( ( G o. card ) ` A )
27		cardf2 8769	. . . . 5 |- card : { x | E. y e. On y ~~ x } --> On
28		ffun 6048	. . . . 5 |- ( card : { x | E. y e. On y ~~ x } --> On -> Fun card )
29	27, 28	ax-mp 5	. . . 4 |- Fun card
30		finnum 8774	. . . 4 |- ( A e. Fin -> A e. dom card )
31		fvco 6274	. . . 4 |- ( ( Fun card /\ A e. dom card ) -> ( ( G o. card ) ` A ) = ( G ` ( card ` A ) ) )
32	29, 30, 31	sylancr 695	. . 3 |- ( A e. Fin -> ( ( G o. card ) ` A ) = ( G ` ( card ` A ) ) )
33	26, 32	syl5eq 2668	. 2 |- ( A e. Fin -> ( ( # |` Fin ) ` A ) = ( G ` ( card ` A ) ) )
34		fvres 6207	. 2 |- ( A e. Fin -> ( ( # |` Fin ) ` A ) = ( # ` A ) )
35	33, 34	eqtr3d 2658	1 |- ( A e. Fin -> ( G ` ( card ` A ) ) = ( # ` A ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   |` cres 5116   o. ccom 5118   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   reccrdg 7505   ~~ cen 7952   Fincfn 7955   cardccrd 8761   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   NN0cn0 11292   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-hash 13118

This theorem is referenced by:  hashginv  13121  hashfz1  13134  hashen  13135  hashcard  13146  hashcl  13147  hashgval2  13167  hashdom  13168  hashun  13171  fz1isolem  13245