Theorem f1dmvrnfibi 7072

Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 7073. (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 5507	. . . 4 |- ( F : A -1-1-> B -> Rel F )
2	1	ad2antlr 489	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ F e. Fin ) -> Rel F )
3		f1cnv 5568	. . . . 5 |- ( F : A -1-1-> B -> `' F : ran F -1-1-onto-> A )
4		f1ofun 5546	. . . . 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 489	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ F e. Fin ) -> Fun `' F )
7		simpr 110	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ F e. Fin ) -> F e. Fin )
8		funrnfi 7070	. . 3 |- ( ( Rel F /\ Fun `' F /\ F e. Fin ) -> ran F e. Fin )
9	2, 6, 7, 8	syl3anc 1250	. 2 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ F e. Fin ) -> ran F e. Fin )
10		simpr 110	. . . 4 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> ran F e. Fin )
11		f1dm 5508	. . . . . . . 8 |- ( F : A -1-1-> B -> dom F = A )
12		f1f1orn 5555	. . . . . . . 8 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )
13		eleq1 2270	. . . . . . . . . . . 12 |- ( A = dom F -> ( A e. V <-> dom F e. V ) )
14		f1oeq2 5533	. . . . . . . . . . . 12 |- ( A = dom F -> ( F : A -1-1-onto-> ran F <-> F : dom F -1-1-onto-> ran F ) )
15	13, 14	anbi12d 473	. . . . . . . . . . 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 2210	. . . . . . . . . 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 144	. . . . . . . . 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 1462	. . . . . . . 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 125	. . . . . 6 |- ( ( A e. V /\ F : A -1-1-> B ) -> ( dom F e. V /\ F : dom F -1-1-onto-> ran F ) )
21	20	adantr 276	. . . . 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 6871	. . . . 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 6997	. . . 4 |- ( ( ran F e. Fin /\ dom F ~~ ran F ) -> dom F e. Fin )
25	10, 23, 24	syl2anc 411	. . 3 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> dom F e. Fin )
26		f1fun 5506	. . . . 5 |- ( F : A -1-1-> B -> Fun F )
27	26	ad2antlr 489	. . . 4 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> Fun F )
28		fundmfibi 7066	. . . 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 167	. 2 |- ( ( ( A e. V /\ F : A -1-1-> B ) /\ ran F e. Fin ) -> F e. Fin )
31	9, 30	impbida 596	1 |- ( ( A e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   class class class wbr 4059   `'ccnv 4692   dom cdm 4693   ran crn 4694   Rel wrel 4698   Fun wfun 5284   -1-1->wf1 5287   -1-1-onto->wf1o 5289   ~~ cen 6848   Fincfn 6850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1st 6249  df-2nd 6250  df-1o 6525  df-er 6643  df-en 6851  df-fin 6853

This theorem is referenced by:  f1vrnfibi  7073