Theorem f1dmvrnfibi 6881

Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6882. (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 5376	. . . 4 |- ( F : A -1-1-> B -> Rel F )
2	1	ad2antlr 481	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ F e. Fin ) -> Rel F )
3		f1cnv 5435	. . . . 5 |- ( F : A -1-1-> B -> `' F : ran F -1-1-onto-> A )
4		f1ofun 5413	. . . . 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 481	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ F e. Fin ) -> Fun `' F )
7		simpr 109	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ F e. Fin ) -> F e. Fin )
8		funrnfi 6879	. . 3 |- ( ( Rel F /\ Fun `' F /\ F e. Fin ) -> ran F e. Fin )
9	2, 6, 7, 8	syl3anc 1220	. 2 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ F e. Fin ) -> ran F e. Fin )
10		simpr 109	. . . 4 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> ran F e. Fin )
11		f1dm 5377	. . . . . . . 8 |- ( F : A -1-1-> B -> dom F = A )
12		f1f1orn 5422	. . . . . . . 8 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )
13		eleq1 2220	. . . . . . . . . . . 12 |- ( A = dom F -> ( A e. V <-> dom F e. V ) )
14		f1oeq2 5401	. . . . . . . . . . . 12 |- ( A = dom F -> ( F : A -1-1-onto-> ran F <-> F : dom F -1-1-onto-> ran F ) )
15	13, 14	anbi12d 465	. . . . . . . . . . 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 2160	. . . . . . . . . 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 143	. . . . . . . . 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 1421	. . . . . . . 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 62	. . . . . . 7 |- ( F : A -1-1-> B -> ( A e. V -> ( dom F e. V /\ F : dom F -1-1-onto-> ran F ) ) )
20	19	impcom 124	. . . . . 6 |- ( ( A e. V /\ F : A -1-1-> B ) -> ( dom F e. V /\ F : dom F -1-1-onto-> ran F ) )
21	20	adantr 274	. . . . 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 6695	. . . . 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 6812	. . . 4 |- ( ( ran F e. Fin /\ dom F ~~ ran F ) -> dom F e. Fin )
25	10, 23, 24	syl2anc 409	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> dom F e. Fin )
26		f1fun 5375	. . . . 5 |- ( F : A -1-1-> B -> Fun F )
27	26	ad2antlr 481	. . . 4 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> Fun F )
28		fundmfibi 6876	. . . 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 166	. 2 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> F e. Fin )
31	9, 30	impbida 586	1 |- ( ( A e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   class class class wbr 3965   `'ccnv 4582   dom cdm 4583   ran crn 4584   Rel wrel 4588   Fun wfun 5161   -1-1->wf1 5164   -1-1-onto->wf1o 5166   ~~ cen 6676   Fincfn 6678

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-iord 4325  df-on 4327  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-1st 6082  df-2nd 6083  df-1o 6357  df-er 6473  df-en 6679  df-fin 6681

This theorem is referenced by:  f1vrnfibi  6882