Theorem f1dmvrnfibi 6578

Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6579. (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 5168	. . . 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 5225	. . . . 5 |- ( F : A -1-1-> B -> `' F : ran F -1-1-onto-> A )
4		f1ofun 5203	. . . . 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 6576	. . 3 |- ( ( Rel F /\ Fun `' F /\ F e. Fin ) -> ran F e. Fin )
9	2, 6, 7, 8	syl3anc 1170	. 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 5169	. . . . . . . 8 |- ( F : A -1-1-> B -> dom F = A )
12		f1f1orn 5212	. . . . . . . 8 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )
13		eleq1 2145	. . . . . . . . . . . 12 |- ( A = dom F -> ( A e. V <-> dom F e. V ) )
14		f1oeq2 5193	. . . . . . . . . . . 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 2086	. . . . . . . . . 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 1371	. . . . . . . 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 6404	. . . . 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 6520	. . . 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 5167	. . . . 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 6573	. . . 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 561	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 1285   e. wcel 1434   class class class wbr 3811   `'ccnv 4400   dom cdm 4401   ran crn 4402   Rel wrel 4406   Fun wfun 4963   -1-1->wf1 4966   -1-1-onto->wf1o 4968   ~~ cen 6385   Fincfn 6387

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-1st 5846  df-2nd 5847  df-1o 6113  df-er 6222  df-en 6388  df-fin 6390

This theorem is referenced by:  f1vrnfibi  6579