Theorem f1finf1o 6912

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 5394	. . . . . 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 5395	. . . . . . . . 9 |- ( F : A -1-1-> B -> F Fn A )
6		fnima 5306	. . . . . . . . 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 3162	. . . . . . . . 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 6840	. . . . . . . . 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 6758	. . . . . . . 8 |- ( ( F : A -1-1-> B /\ A C_ A /\ A e. Fin ) -> ( F " A ) ~~ A )
15	1, 10, 13, 14	syl3anc 1228	. . . . . . 7 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( F " A ) ~~ A )
16	8, 15	eqbrtrrd 4006	. . . . . 6 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F ~~ A )
17		entr 6750	. . . . . 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 6897	. . . . 5 |- ( ( B e. Fin /\ ran F C_ B /\ ran F ~~ B ) -> ran F = B )
20	2, 4, 18, 19	syl3anc 1228	. . . 4 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F = B )
21		dff1o5 5441	. . . 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 5431	. 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 1343   e. wcel 2136   C_ wss 3116   class class class wbr 3982   ran crn 4605   "cima 4607   Fn wfn 5183   -1-1->wf1 5185   -1-1-onto->wf1o 5187   ~~ cen 6704   Fincfn 6706

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1o 6384  df-er 6501  df-en 6707  df-fin 6709

This theorem is referenced by:  iseqf1olemqf1o  10428  crth  12156  eulerthlemh  12163  pwf1oexmid  13879