Theorem f1finf1o 6936

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 5414	. . . . . 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 5415	. . . . . . . . 9 |- ( F : A -1-1-> B -> F Fn A )
6		fnima 5326	. . . . . . . . 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 3173	. . . . . . . . 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 6864	. . . . . . . . 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 6782	. . . . . . . 8 |- ( ( F : A -1-1-> B /\ A C_ A /\ A e. Fin ) -> ( F " A ) ~~ A )
15	1, 10, 13, 14	syl3anc 1238	. . . . . . 7 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( F " A ) ~~ A )
16	8, 15	eqbrtrrd 4022	. . . . . 6 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F ~~ A )
17		entr 6774	. . . . . 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 6921	. . . . 5 |- ( ( B e. Fin /\ ran F C_ B /\ ran F ~~ B ) -> ran F = B )
20	2, 4, 18, 19	syl3anc 1238	. . . 4 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F = B )
21		dff1o5 5462	. . . 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 5452	. 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 1353   e. wcel 2146   C_ wss 3127   class class class wbr 3998   ran crn 4621   "cima 4623   Fn wfn 5203   -1-1->wf1 5205   -1-1-onto->wf1o 5207   ~~ cen 6728   Fincfn 6730

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1o 6407  df-er 6525  df-en 6731  df-fin 6733

This theorem is referenced by:  iseqf1olemqf1o  10463  crth  12191  eulerthlemh  12198  pwf1oexmid  14318