Theorem fihashf1rn 10567

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 5338	. . 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 6833	. . 3 |- ( ( F Fn A /\ A e. Fin ) -> F e. Fin )
4	1, 2, 3	syl2an2 584	. 2 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> F e. Fin )
5		f1o2ndf1 6133	. . . 4 |- ( F : A -1-1-> B -> ( 2nd |` F ) : F -1-1-onto-> ran F )
6		df-2nd 6047	. . . . . . . . 9 |- 2nd = ( x e. _V |-> U. ran { x } )
7	6	funmpt2 5170	. . . . . . . 8 |- Fun 2nd
8		f1f 5336	. . . . . . . . . . 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 5655	. . . . . . . . 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 5649	. . . . . . . 8 |- ( ( Fun 2nd /\ F e. _V ) -> ( 2nd |` F ) e. _V )
14	7, 12, 13	sylancr 411	. . . . . . 7 |- ( ( A e. Fin /\ F : A -1-1-> B ) -> ( 2nd |` F ) e. _V )
15		f1oeq1 5364	. . . . . . . . . 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 2143	. . . . . . . 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 2775	. . . . . 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 10566	. . . 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 1792	. 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 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689   {csn 3532   U.cuni 3744   ran crn 4548   |` cres 4549   Fun wfun 5125   Fn wfn 5126   -->wf 5127   -1-1->wf1 5128   -1-1-onto->wf1o 5130   ` cfv 5131   2ndc2nd 6045   Fincfn 6642  ♯chash 10553

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-2nd 6047  df-recs 6210  df-frec 6296  df-1o 6321  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-ihash 10554

This theorem is referenced by: (None)