Theorem hashen 11092

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 6977	. . . 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 6977	. . . . 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 537	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n e. om )
8		simprl 531	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m e. om )
9		nneneq 7086	. . . . 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 538	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> A ~~ n )
12		enen1 7069	. . . . . 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 533	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> B ~~ m )
15		enen2 7070	. . . . . 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 7000	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n ~~ A )
19		hashennn 11088	. . . . . . 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 7000	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m ~~ B )
22		hashennn 11088	. . . . . . 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 2246	. . . . 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 9535	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> 0 e. ZZ )
26		eqid 2231	. . . . . . . 8 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
27	25, 26	frec2uzf1od 10714	. . . . . . 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 5591	. . . . . . 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 5923	. . . . . 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 1272	. . . . 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 2656	. 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> ( (♯ ` A ) = (♯ ` B ) <-> A ~~ B ) )
35	3, 34	rexlimddv 2656	1 |- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) = (♯ ` B ) <-> A ~~ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   E.wrex 2512   class class class wbr 4093   |-> cmpt 4155   omcom 4694   -1-1->wf1 5330   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028  freccfrec 6599   ~~ cen 6950   Fincfn 6952   0cc0 8075   1c1 8076   + caddc 8078   ZZcz 9523   ZZ>=cuz 9799  ♯chash 11083

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-recs 6514  df-frec 6600  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-ihash 11084

This theorem is referenced by:  hasheqf1o  11093  isfinite4im  11100  fihasheq0  11101  hashsng  11106  fihashen1  11107  fihashfn  11109  hashun  11114  hashfz  11131  hashxp  11136  hash2en  11153  mertenslemi1  12159  hashdvds  12856  crth  12859  phimullem  12860  eulerth  12868  4sqlem11  13037  znhash  14735  lgsquadlem1  15879  lgsquadlem2  15880  lgsquadlem3  15881