Theorem f1finf1o 6843

Description: Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.)

Assertion

Ref	Expression
f1finf1o	|- ( ( A ~~ B /\ B e. Fin ) -> ( F : A -1-1-> B <-> F : A -1-1-onto-> B ) )

Proof of Theorem f1finf1o

Step	Hyp	Ref	Expression
1		simpr 109	. . . 4 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> F : A -1-1-> B )
2		simplr 520	. . . . 5 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> B e. Fin )
3		f1rn 5337	. . . . . 6 |- ( F : A -1-1-> B -> ran F C_ B )
4	3	adantl 275	. . . . 5 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F C_ B )
5		f1fn 5338	. . . . . . . . 9 |- ( F : A -1-1-> B -> F Fn A )
6		fnima 5249	. . . . . . . . 9 |- ( F Fn A -> ( F " A ) = ran F )
7	5, 6	syl 14	. . . . . . . 8 |- ( F : A -1-1-> B -> ( F " A ) = ran F )
8	7	adantl 275	. . . . . . 7 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( F " A ) = ran F )
9		ssid 3122	. . . . . . . . 9 |- A C_ A
10	9	a1i 9	. . . . . . . 8 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> A C_ A )
11		simpll 519	. . . . . . . . 9 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> A ~~ B )
12		enfii 6776	. . . . . . . . 9 |- ( ( B e. Fin /\ A ~~ B ) -> A e. Fin )
13	2, 11, 12	syl2anc 409	. . . . . . . 8 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> A e. Fin )
14		f1imaeng 6694	. . . . . . . 8 |- ( ( F : A -1-1-> B /\ A C_ A /\ A e. Fin ) -> ( F " A ) ~~ A )
15	1, 10, 13, 14	syl3anc 1217	. . . . . . 7 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( F " A ) ~~ A )
16	8, 15	eqbrtrrd 3960	. . . . . 6 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F ~~ A )
17		entr 6686	. . . . . 6 |- ( ( ran F ~~ A /\ A ~~ B ) -> ran F ~~ B )
18	16, 11, 17	syl2anc 409	. . . . 5 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F ~~ B )
19		fisseneq 6828	. . . . 5 |- ( ( B e. Fin /\ ran F C_ B /\ ran F ~~ B ) -> ran F = B )
20	2, 4, 18, 19	syl3anc 1217	. . . 4 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F = B )
21		dff1o5 5384	. . . 4 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) )
22	1, 20, 21	sylanbrc 414	. . 3 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> F : A -1-1-onto-> B )
23	22	ex 114	. 2 |- ( ( A ~~ B /\ B e. Fin ) -> ( F : A -1-1-> B -> F : A -1-1-onto-> B ) )
24		f1of1 5374	. 2 |- ( F : A -1-1-onto-> B -> F : A -1-1-> B )
25	23, 24	impbid1 141	1 |- ( ( A ~~ B /\ B e. Fin ) -> ( F : A -1-1-> B <-> F : A -1-1-onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   C_ wss 3076   class class class wbr 3937   ran crn 4548   "cima 4550   Fn wfn 5126   -1-1->wf1 5128   -1-1-onto->wf1o 5130   ~~ cen 6640   Fincfn 6642

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

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-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-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-1o 6321  df-er 6437  df-en 6643  df-fin 6645

This theorem is referenced by:  iseqf1olemqf1o  10297  crth  11936  pwf1oexmid  13367