Theorem f1finf1o 7075

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 110	. . . 4 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> F : A -1-1-> B )
2		simplr 528	. . . . 5 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> B e. Fin )
3		f1rn 5504	. . . . . 6 |- ( F : A -1-1-> B -> ran F C_ B )
4	3	adantl 277	. . . . 5 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F C_ B )
5		f1fn 5505	. . . . . . . . 9 |- ( F : A -1-1-> B -> F Fn A )
6		fnima 5414	. . . . . . . . 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 277	. . . . . . 7 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( F " A ) = ran F )
9		ssid 3221	. . . . . . . . 9 |- A C_ A
10	9	a1i 9	. . . . . . . 8 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> A C_ A )
11		simpll 527	. . . . . . . . 9 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> A ~~ B )
12		enfii 6997	. . . . . . . . 9 |- ( ( B e. Fin /\ A ~~ B ) -> A e. Fin )
13	2, 11, 12	syl2anc 411	. . . . . . . 8 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> A e. Fin )
14		f1imaeng 6907	. . . . . . . 8 |- ( ( F : A -1-1-> B /\ A C_ A /\ A e. Fin ) -> ( F " A ) ~~ A )
15	1, 10, 13, 14	syl3anc 1250	. . . . . . 7 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( F " A ) ~~ A )
16	8, 15	eqbrtrrd 4083	. . . . . 6 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F ~~ A )
17		entr 6899	. . . . . 6 |- ( ( ran F ~~ A /\ A ~~ B ) -> ran F ~~ B )
18	16, 11, 17	syl2anc 411	. . . . 5 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F ~~ B )
19		fisseneq 7057	. . . . 5 |- ( ( B e. Fin /\ ran F C_ B /\ ran F ~~ B ) -> ran F = B )
20	2, 4, 18, 19	syl3anc 1250	. . . 4 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F = B )
21		dff1o5 5553	. . . 4 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) )
22	1, 20, 21	sylanbrc 417	. . 3 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> F : A -1-1-onto-> B )
23	22	ex 115	. 2 |- ( ( A ~~ B /\ B e. Fin ) -> ( F : A -1-1-> B -> F : A -1-1-onto-> B ) )
24		f1of1 5543	. 2 |- ( F : A -1-1-onto-> B -> F : A -1-1-> B )
25	23, 24	impbid1 142	1 |- ( ( A ~~ B /\ B e. Fin ) -> ( F : A -1-1-> B <-> F : A -1-1-onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   C_ wss 3174   class class class wbr 4059   ran crn 4694   "cima 4696   Fn wfn 5285   -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-1o 6525  df-er 6643  df-en 6851  df-fin 6853

This theorem is referenced by:  iseqf1olemqf1o  10688  crth  12661  eulerthlemh  12668  lgseisenlem2  15663  pwf1oexmid  16138