Theorem f1dmvrnfibi 6651

Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6652. (Contributed by AV, 10-Jan-2020.)

Assertion

Ref	Expression
f1dmvrnfibi	|- ( ( A e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )

Proof of Theorem f1dmvrnfibi

Step	Hyp	Ref	Expression
1		f1rel 5220	. . . 4 |- ( F : A -1-1-> B -> Rel F )
2	1	ad2antlr 473	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ F e. Fin ) -> Rel F )
3		f1cnv 5277	. . . . 5 |- ( F : A -1-1-> B -> `' F : ran F -1-1-onto-> A )
4		f1ofun 5255	. . . . 5 |- ( `' F : ran F -1-1-onto-> A -> Fun `' F )
5	3, 4	syl 14	. . . 4 |- ( F : A -1-1-> B -> Fun `' F )
6	5	ad2antlr 473	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ F e. Fin ) -> Fun `' F )
7		simpr 108	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ F e. Fin ) -> F e. Fin )
8		funrnfi 6649	. . 3 |- ( ( Rel F /\ Fun `' F /\ F e. Fin ) -> ran F e. Fin )
9	2, 6, 7, 8	syl3anc 1174	. 2 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ F e. Fin ) -> ran F e. Fin )
10		simpr 108	. . . 4 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> ran F e. Fin )
11		f1dm 5221	. . . . . . . 8 |- ( F : A -1-1-> B -> dom F = A )
12		f1f1orn 5264	. . . . . . . 8 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )
13		eleq1 2150	. . . . . . . . . . . 12 |- ( A = dom F -> ( A e. V <-> dom F e. V ) )
14		f1oeq2 5245	. . . . . . . . . . . 12 |- ( A = dom F -> ( F : A -1-1-onto-> ran F <-> F : dom F -1-1-onto-> ran F ) )
15	13, 14	anbi12d 457	. . . . . . . . . . 11 |- ( A = dom F -> ( ( A e. V /\ F : A -1-1-onto-> ran F ) <-> ( dom F e. V /\ F : dom F -1-1-onto-> ran F ) ) )
16	15	eqcoms 2091	. . . . . . . . . 10 |- ( dom F = A -> ( ( A e. V /\ F : A -1-1-onto-> ran F ) <-> ( dom F e. V /\ F : dom F -1-1-onto-> ran F ) ) )
17	16	biimpd 142	. . . . . . . . 9 |- ( dom F = A -> ( ( A e. V /\ F : A -1-1-onto-> ran F ) -> ( dom F e. V /\ F : dom F -1-1-onto-> ran F ) ) )
18	17	expcomd 1375	. . . . . . . 8 |- ( dom F = A -> ( F : A -1-1-onto-> ran F -> ( A e. V -> ( dom F e. V /\ F : dom F -1-1-onto-> ran F ) ) ) )
19	11, 12, 18	sylc 61	. . . . . . 7 |- ( F : A -1-1-> B -> ( A e. V -> ( dom F e. V /\ F : dom F -1-1-onto-> ran F ) ) )
20	19	impcom 123	. . . . . 6 |- ( ( A e. V /\ F : A -1-1-> B ) -> ( dom F e. V /\ F : dom F -1-1-onto-> ran F ) )
21	20	adantr 270	. . . . 5 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> ( dom F e. V /\ F : dom F -1-1-onto-> ran F ) )
22		f1oeng 6472	. . . . 5 |- ( ( dom F e. V /\ F : dom F -1-1-onto-> ran F ) -> dom F ~~ ran F )
23	21, 22	syl 14	. . . 4 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> dom F ~~ ran F )
24		enfii 6588	. . . 4 |- ( ( ran F e. Fin /\ dom F ~~ ran F ) -> dom F e. Fin )
25	10, 23, 24	syl2anc 403	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> dom F e. Fin )
26		f1fun 5219	. . . . 5 |- ( F : A -1-1-> B -> Fun F )
27	26	ad2antlr 473	. . . 4 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> Fun F )
28		fundmfibi 6646	. . . 4 |- ( Fun F -> ( F e. Fin <-> dom F e. Fin ) )
29	27, 28	syl 14	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> ( F e. Fin <-> dom F e. Fin ) )
30	25, 29	mpbird 165	. 2 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> F e. Fin )
31	9, 30	impbida 563	1 |- ( ( A e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   class class class wbr 3845   `'ccnv 4437   dom cdm 4438   ran crn 4439   Rel wrel 4443   Fun wfun 5009   -1-1->wf1 5012   -1-1-onto->wf1o 5014   ~~ cen 6453   Fincfn 6455

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912  df-1o 6181  df-er 6290  df-en 6456  df-fin 6458

This theorem is referenced by:  f1vrnfibi  6652