Theorem fihashf1rn 11010

Description: The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.)

Assertion

Ref	Expression
fihashf1rn	|- ( ( A e. Fin /\ F : A -1-1-> B ) -> (♯ ` F ) = (♯ ` ran F ) )

Proof of Theorem fihashf1rn

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1fn 5533	. . 3 |- ( F : A -1-1-> B -> F Fn A )
2		simpl 109	. . 3 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> A e. Fin )
3		fnfi 7103	. . 3 |- ( ( F Fn A /\ A e. Fin ) -> F e. Fin )
4	1, 2, 3	syl2an2 596	. 2 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> F e. Fin )
5		f1o2ndf1 6374	. . . 4 |- ( F : A -1-1-> B -> ( 2nd |` F ) : F -1-1-onto-> ran F )
6		df-2nd 6287	. . . . . . . . 9 |- 2nd = ( x e. _V |-> U. ran { x } )
7	6	funmpt2 5357	. . . . . . . 8 |- Fun 2nd
8		f1f 5531	. . . . . . . . . . 11 |- ( F : A -1-1-> B -> F : A --> B )
9	8	anim2i 342	. . . . . . . . . 10 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> ( A e. Fin /\ F : A --> B ) )
10	9	ancomd 267	. . . . . . . . 9 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> ( F : A --> B /\ A e. Fin ) )
11		fex 5868	. . . . . . . . 9 |- ( ( F : A --> B /\ A e. Fin ) -> F e. _V )
12	10, 11	syl 14	. . . . . . . 8 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> F e. _V )
13		resfunexg 5860	. . . . . . . 8 |- ( ( Fun 2nd /\ F e. _V ) -> ( 2nd |` F ) e. _V )
14	7, 12, 13	sylancr 414	. . . . . . 7 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> ( 2nd |` F ) e. _V )
15		f1oeq1 5560	. . . . . . . . . 10 |- ( ( 2nd |` F ) = f -> ( ( 2nd |` F ) : F -1-1-onto-> ran F <-> f : F -1-1-onto-> ran F ) )
16	15	biimpd 144	. . . . . . . . 9 |- ( ( 2nd |` F ) = f -> ( ( 2nd |` F ) : F -1-1-onto-> ran F -> f : F -1-1-onto-> ran F ) )
17	16	eqcoms 2232	. . . . . . . 8 |- ( f = ( 2nd |` F ) -> ( ( 2nd |` F ) : F -1-1-onto-> ran F -> f : F -1-1-onto-> ran F ) )
18	17	adantl 277	. . . . . . 7 |- ( ( ( A e. Fin /\ F : A -1-1-> B ) /\ f = ( 2nd |` F ) ) -> ( ( 2nd |` F ) : F -1-1-onto-> ran F -> f : F -1-1-onto-> ran F ) )
19	14, 18	spcimedv 2889	. . . . . 6 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> ( ( 2nd |` F ) : F -1-1-onto-> ran F -> E. f f : F -1-1-onto-> ran F ) )
20	19	ex 115	. . . . 5 |- ( A e. Fin -> ( F : A -1-1-> B -> ( ( 2nd |` F ) : F -1-1-onto-> ran F -> E. f f : F -1-1-onto-> ran F ) ) )
21	20	com13 80	. . . 4 |- ( ( 2nd |` F ) : F -1-1-onto-> ran F -> ( F : A -1-1-> B -> ( A e. Fin -> E. f f : F -1-1-onto-> ran F ) ) )
22	5, 21	mpcom 36	. . 3 |- ( F : A -1-1-> B -> ( A e. Fin -> E. f f : F -1-1-onto-> ran F ) )
23	22	impcom 125	. 2 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> E. f f : F -1-1-onto-> ran F )
24		fihasheqf1oi 11009	. . . 4 |- ( ( F e. Fin /\ f : F -1-1-onto-> ran F ) -> (♯ ` F ) = (♯ ` ran F ) )
25	24	ex 115	. . 3 |- ( F e. Fin -> ( f : F -1-1-onto-> ran F -> (♯ ` F ) = (♯ ` ran F ) ) )
26	25	exlimdv 1865	. 2 |- ( F e. Fin -> ( E. f f : F -1-1-onto-> ran F -> (♯ ` F ) = (♯ ` ran F ) ) )
27	4, 23, 26	sylc 62	1 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> (♯ ` F ) = (♯ ` ran F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   {csn 3666   U.cuni 3888   ran crn 4720   |` cres 4721   Fun wfun 5312   Fn wfn 5313   -->wf 5314   -1-1->wf1 5315   -1-1-onto->wf1o 5317   ` cfv 5318   2ndc2nd 6285   Fincfn 6887  ♯chash 10997

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-ihash 10998

This theorem is referenced by: (None)