Theorem hashen 10910

Description: Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

Assertion

Ref	Expression
hashen	|- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) = (♯ ` B ) <-> A ~~ B ) )

Proof of Theorem hashen

Dummy variables m n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfi 6838	. . . 4 |- ( A e. Fin <-> E. n e. om A ~~ n )
2	1	biimpi 120	. . 3 |- ( A e. Fin -> E. n e. om A ~~ n )
3	2	adantr 276	. 2 |- ( ( A e. Fin /\ B e. Fin ) -> E. n e. om A ~~ n )
4		isfi 6838	. . . . 5 |- ( B e. Fin <-> E. m e. om B ~~ m )
5	4	biimpi 120	. . . 4 |- ( B e. Fin -> E. m e. om B ~~ m )
6	5	ad2antlr 489	. . 3 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om B ~~ m )
7		simplrl 535	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n e. om )
8		simprl 529	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m e. om )
9		nneneq 6936	. . . . 5 |- ( ( n e. om /\ m e. om ) -> ( n ~~ m <-> n = m ) )
10	7, 8, 9	syl2anc 411	. . . 4 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( n ~~ m <-> n = m ) )
11		simplrr 536	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> A ~~ n )
12		enen1 6919	. . . . . 6 |- ( A ~~ n -> ( A ~~ B <-> n ~~ B ) )
13	11, 12	syl 14	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( A ~~ B <-> n ~~ B ) )
14		simprr 531	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> B ~~ m )
15		enen2 6920	. . . . . 6 |- ( B ~~ m -> ( n ~~ B <-> n ~~ m ) )
16	14, 15	syl 14	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( n ~~ B <-> n ~~ m ) )
17	13, 16	bitrd 188	. . . 4 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( A ~~ B <-> n ~~ m ) )
18	11	ensymd 6860	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n ~~ A )
19		hashennn 10906	. . . . . . 7 |- ( ( n e. om /\ n ~~ A ) -> (♯ ` A ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) )
20	7, 18, 19	syl2anc 411	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> (♯ ` A ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) )
21	14	ensymd 6860	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m ~~ B )
22		hashennn 10906	. . . . . . 7 |- ( ( m e. om /\ m ~~ B ) -> (♯ ` B ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) )
23	8, 21, 22	syl2anc 411	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> (♯ ` B ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) )
24	20, 23	eqeq12d 2219	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( (♯ ` A ) = (♯ ` B ) <-> (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) ) )
25		0zd 9366	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> 0 e. ZZ )
26		eqid 2204	. . . . . . . 8 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
27	25, 26	frec2uzf1od 10532	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) : om -1-1-onto-> ( ZZ>= ` 0 ) )
28		f1of1 5515	. . . . . . 7 |- (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) : om -1-1-onto-> ( ZZ>= ` 0 ) -> frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) : om -1-1-> ( ZZ>= ` 0 ) )
29	27, 28	syl 14	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) : om -1-1-> ( ZZ>= ` 0 ) )
30		f1fveq 5831	. . . . . 6 |- ( (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) : om -1-1-> ( ZZ>= ` 0 ) /\ ( n e. om /\ m e. om ) ) -> ( (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) <-> n = m ) )
31	29, 7, 8, 30	syl12anc 1247	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) <-> n = m ) )
32	24, 31	bitrd 188	. . . 4 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( (♯ ` A ) = (♯ ` B ) <-> n = m ) )
33	10, 17, 32	3bitr4rd 221	. . 3 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( (♯ ` A ) = (♯ ` B ) <-> A ~~ B ) )
34	6, 33	rexlimddv 2627	. 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> ( (♯ ` A ) = (♯ ` B ) <-> A ~~ B ) )
35	3, 34	rexlimddv 2627	1 |- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) = (♯ ` B ) <-> A ~~ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   E.wrex 2484   class class class wbr 4043   |-> cmpt 4104   omcom 4636   -1-1->wf1 5265   -1-1-onto->wf1o 5267   ` cfv 5268  (class class class)co 5934  freccfrec 6466   ~~ cen 6815   Fincfn 6817   0cc0 7907   1c1 7908   + caddc 7910   ZZcz 9354   ZZ>=cuz 9630  ♯chash 10901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-addcom 8007  ax-addass 8009  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-0id 8015  ax-rnegex 8016  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-ltwlin 8020  ax-pre-lttrn 8021  ax-pre-ltadd 8023

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4338  df-iord 4411  df-on 4413  df-ilim 4414  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-recs 6381  df-frec 6467  df-er 6610  df-en 6818  df-dom 6819  df-fin 6820  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-inn 9019  df-n0 9278  df-z 9355  df-uz 9631  df-ihash 10902

This theorem is referenced by:  hasheqf1o  10911  isfinite4im  10918  fihasheq0  10919  hashsng  10924  fihashen1  10925  fihashfn  10926  hashun  10931  hashfz  10947  hashxp  10952  mertenslemi1  11765  hashdvds  12462  crth  12465  phimullem  12466  eulerth  12474  4sqlem11  12643  znhash  14336  lgsquadlem1  15472  lgsquadlem2  15473  lgsquadlem3  15474