Theorem hashen 10927

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 6851	. . . 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 6851	. . . . 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 6953	. . . . 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 6936	. . . . . 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 6937	. . . . . 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 6874	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n ~~ A )
19		hashennn 10923	. . . . . . 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 6874	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m ~~ B )
22		hashennn 10923	. . . . . . 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 9383	. . . . . . . 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 10549	. . . . . . 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 5520	. . . . . . 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 5840	. . . . . 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 4637   -1-1->wf1 5267   -1-1-onto->wf1o 5269   ` cfv 5270  (class class class)co 5943  freccfrec 6475   ~~ cen 6824   Fincfn 6826   0cc0 7924   1c1 7925   + caddc 7927   ZZcz 9371   ZZ>=cuz 9647  ♯chash 10918

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 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-ltadd 8040

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 4339  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-recs 6390  df-frec 6476  df-er 6619  df-en 6827  df-dom 6828  df-fin 6829  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-inn 9036  df-n0 9295  df-z 9372  df-uz 9648  df-ihash 10919

This theorem is referenced by:  hasheqf1o  10928  isfinite4im  10935  fihasheq0  10936  hashsng  10941  fihashen1  10942  fihashfn  10943  hashun  10948  hashfz  10964  hashxp  10969  hash2en  10986  mertenslemi1  11788  hashdvds  12485  crth  12488  phimullem  12489  eulerth  12497  4sqlem11  12666  znhash  14360  lgsquadlem1  15496  lgsquadlem2  15497  lgsquadlem3  15498