Theorem fihashf1rn 11227

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 5600	. . 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 7250	. . 3 |- ( ( F Fn A /\ A e. Fin ) -> F e. Fin )
4	1, 2, 3	syl2an2 602	. 2 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> F e. Fin )
5		f1o2ndf1 6464	. . . 4 |- ( F : A -1-1-> B -> ( 2nd |` F ) : F -1-1-onto-> ran F )
6		df-2nd 6375	. . . . . . . . 9 |- 2nd = ( x e. _V |-> U. ran { x } )
7	6	funmpt2 5416	. . . . . . . 8 |- Fun 2nd
8		f1f 5598	. . . . . . . . . . 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 5947	. . . . . . . . 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 5936	. . . . . . . 8 |- ( ( Fun 2nd /\ F e. _V ) -> ( 2nd |` F ) e. _V )
14	7, 12, 13	sylancr 418	. . . . . . 7 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> ( 2nd |` F ) e. _V )
15		f1oeq1 5627	. . . . . . . . . 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 2241	. . . . . . . 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 2911	. . . . . 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 11226	. . . 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 1872	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   {csn 3709   U.cuni 3935   ran crn 4775   |` cres 4776   Fun wfun 5371   Fn wfn 5372   -->wf 5373   -1-1->wf1 5374   -1-1-onto->wf1o 5376   ` cfv 5377   2ndc2nd 6373   Fincfn 7022  ♯chash 11214

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-ihash 11215

This theorem is used by: (None)