Theorem hashen 10893

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 6829	. . . 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 6829	. . . . 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 6927	. . . . 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 6910	. . . . . 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 6911	. . . . . 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 6851	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n ~~ A )
19		hashennn 10889	. . . . . . 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 6851	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m ~~ B )
22		hashennn 10889	. . . . . . 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 2211	. . . . 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 9355	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> 0 e. ZZ )
26		eqid 2196	. . . . . . . 8 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
27	25, 26	frec2uzf1od 10515	. . . . . . 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 5506	. . . . . . 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 5822	. . . . . 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 2619	. 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> ( (♯ ` A ) = (♯ ` B ) <-> A ~~ B ) )
35	3, 34	rexlimddv 2619	1 |- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) = (♯ ` B ) <-> A ~~ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   E.wrex 2476   class class class wbr 4034   |-> cmpt 4095   omcom 4627   -1-1->wf1 5256   -1-1-onto->wf1o 5258   ` cfv 5259  (class class class)co 5925  freccfrec 6457   ~~ cen 6806   Fincfn 6808   0cc0 7896   1c1 7897   + caddc 7899   ZZcz 9343   ZZ>=cuz 9618  ♯chash 10884

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-recs 6372  df-frec 6458  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-ihash 10885

This theorem is referenced by:  hasheqf1o  10894  isfinite4im  10901  fihasheq0  10902  hashsng  10907  fihashen1  10908  fihashfn  10909  hashun  10914  hashfz  10930  hashxp  10935  mertenslemi1  11717  hashdvds  12414  crth  12417  phimullem  12418  eulerth  12426  4sqlem11  12595  znhash  14288  lgsquadlem1  15402  lgsquadlem2  15403  lgsquadlem3  15404