Theorem fihashf1rn 10723

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 5405	. . 3 |- ( F : A -1-1-> B -> F Fn A )
2		simpl 108	. . 3 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> A e. Fin )
3		fnfi 6914	. . 3 |- ( ( F Fn A /\ A e. Fin ) -> F e. Fin )
4	1, 2, 3	syl2an2 589	. 2 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> F e. Fin )
5		f1o2ndf1 6207	. . . 4 |- ( F : A -1-1-> B -> ( 2nd |` F ) : F -1-1-onto-> ran F )
6		df-2nd 6120	. . . . . . . . 9 |- 2nd = ( x e. _V |-> U. ran { x } )
7	6	funmpt2 5237	. . . . . . . 8 |- Fun 2nd
8		f1f 5403	. . . . . . . . . . 11 |- ( F : A -1-1-> B -> F : A --> B )
9	8	anim2i 340	. . . . . . . . . 10 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> ( A e. Fin /\ F : A --> B ) )
10	9	ancomd 265	. . . . . . . . 9 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> ( F : A --> B /\ A e. Fin ) )
11		fex 5725	. . . . . . . . 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 5717	. . . . . . . 8 |- ( ( Fun 2nd /\ F e. _V ) -> ( 2nd |` F ) e. _V )
14	7, 12, 13	sylancr 412	. . . . . . 7 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> ( 2nd |` F ) e. _V )
15		f1oeq1 5431	. . . . . . . . . 10 |- ( ( 2nd |` F ) = f -> ( ( 2nd |` F ) : F -1-1-onto-> ran F <-> f : F -1-1-onto-> ran F ) )
16	15	biimpd 143	. . . . . . . . 9 |- ( ( 2nd |` F ) = f -> ( ( 2nd |` F ) : F -1-1-onto-> ran F -> f : F -1-1-onto-> ran F ) )
17	16	eqcoms 2173	. . . . . . . 8 |- ( f = ( 2nd |` F ) -> ( ( 2nd |` F ) : F -1-1-onto-> ran F -> f : F -1-1-onto-> ran F ) )
18	17	adantl 275	. . . . . . 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 2816	. . . . . 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 114	. . . . 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 124	. 2 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> E. f f : F -1-1-onto-> ran F )
24		fihasheqf1oi 10722	. . . 4 |- ( ( F e. Fin /\ f : F -1-1-onto-> ran F ) -> (♯ ` F ) = (♯ ` ran F ) )
25	24	ex 114	. . 3 |- ( F e. Fin -> ( f : F -1-1-onto-> ran F -> (♯ ` F ) = (♯ ` ran F ) ) )
26	25	exlimdv 1812	. 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 103   = wceq 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730   {csn 3583   U.cuni 3796   ran crn 4612   |` cres 4613   Fun wfun 5192   Fn wfn 5193   -->wf 5194   -1-1->wf1 5195   -1-1-onto->wf1o 5197   ` cfv 5198   2ndc2nd 6118   Fincfn 6718  ♯chash 10709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-ihash 10710

This theorem is referenced by: (None)