Theorem hashen 10968

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 6877	. . . 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 6877	. . . . 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 6981	. . . . 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 6964	. . . . . 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 6965	. . . . . 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 6900	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n ~~ A )
19		hashennn 10964	. . . . . . 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 6900	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m ~~ B )
22		hashennn 10964	. . . . . . 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 2222	. . . . 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 9421	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> 0 e. ZZ )
26		eqid 2207	. . . . . . . 8 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
27	25, 26	frec2uzf1od 10590	. . . . . . 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 5544	. . . . . . 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 5866	. . . . . 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 1248	. . . . 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 2631	. 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> ( (♯ ` A ) = (♯ ` B ) <-> A ~~ B ) )
35	3, 34	rexlimddv 2631	1 |- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) = (♯ ` B ) <-> A ~~ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   E.wrex 2487   class class class wbr 4060   |-> cmpt 4122   omcom 4657   -1-1->wf1 5288   -1-1-onto->wf1o 5290   ` cfv 5291  (class class class)co 5969  freccfrec 6501   ~~ cen 6850   Fincfn 6852   0cc0 7962   1c1 7963   + caddc 7965   ZZcz 9409   ZZ>=cuz 9685  ♯chash 10959

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4176  ax-sep 4179  ax-nul 4187  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-iinf 4655  ax-cnex 8053  ax-resscn 8054  ax-1cn 8055  ax-1re 8056  ax-icn 8057  ax-addcl 8058  ax-addrcl 8059  ax-mulcl 8060  ax-addcom 8062  ax-addass 8064  ax-distr 8066  ax-i2m1 8067  ax-0lt1 8068  ax-0id 8070  ax-rnegex 8071  ax-cnre 8073  ax-pre-ltirr 8074  ax-pre-ltwlin 8075  ax-pre-lttrn 8076  ax-pre-ltadd 8078

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-nul 3470  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-iun 3944  df-br 4061  df-opab 4123  df-mpt 4124  df-tr 4160  df-id 4359  df-iord 4432  df-on 4434  df-ilim 4435  df-suc 4437  df-iom 4658  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-f1o 5298  df-fv 5299  df-riota 5924  df-ov 5972  df-oprab 5973  df-mpo 5974  df-recs 6416  df-frec 6502  df-er 6645  df-en 6853  df-dom 6854  df-fin 6855  df-pnf 8146  df-mnf 8147  df-xr 8148  df-ltxr 8149  df-le 8150  df-sub 8282  df-neg 8283  df-inn 9074  df-n0 9333  df-z 9410  df-uz 9686  df-ihash 10960

This theorem is referenced by:  hasheqf1o  10969  isfinite4im  10976  fihasheq0  10977  hashsng  10982  fihashen1  10983  fihashfn  10984  hashun  10989  hashfz  11005  hashxp  11010  hash2en  11027  mertenslemi1  12007  hashdvds  12704  crth  12707  phimullem  12708  eulerth  12716  4sqlem11  12885  znhash  14579  lgsquadlem1  15715  lgsquadlem2  15716  lgsquadlem3  15717