Metamath Proof Explorer

Theorem f1finf1o

Description: Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010) (Revised by Mario Carneiro, 27-Feb-2014)

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

Proof

Step Hyp Ref Expression
1 simpr 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> F : A -1-1-> B )
2 f1f 
 |-  ( F : A -1-1-> B -> F : A --> B )
3 2 adantl 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> F : A --> B )
4 3 ffnd 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> F Fn A )
5 simpll 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> A ~~ B )
6 3 frnd 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F C_ B )
7 df-pss 
 |-  ( ran F C. B <-> ( ran F C_ B /\ ran F =/= B ) )
8 7 baib 
 |-  ( ran F C_ B -> ( ran F C. B <-> ran F =/= B ) )
9 6 8 syl 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( ran F C. B <-> ran F =/= B ) )
10 simplr 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> B e. Fin )
11 relen 
 |-  Rel ~~
12 11 brrelex1i 
 |-  ( A ~~ B -> A e. _V )
13 5 12 syl 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> A e. _V )
14 10 13 elmapd 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( F e. ( B ^m A ) <-> F : A --> B ) )
15 3 14 mpbird 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> F e. ( B ^m A ) )
16 f1f1orn 
 |-  ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )
17 16 adantl 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> F : A -1-1-onto-> ran F )
18 f1oen3g 
 |-  ( ( F e. ( B ^m A ) /\ F : A -1-1-onto-> ran F ) -> A ~~ ran F )
19 15 17 18 syl2anc 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> A ~~ ran F )
20 php3 
 |-  ( ( B e. Fin /\ ran F C. B ) -> ran F ~< B )
21 20 ex 
 |-  ( B e. Fin -> ( ran F C. B -> ran F ~< B ) )
22 10 21 syl 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( ran F C. B -> ran F ~< B ) )
23 ensdomtr 
 |-  ( ( A ~~ ran F /\ ran F ~< B ) -> A ~< B )
24 19 22 23 syl6an 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( ran F C. B -> A ~< B ) )
25 sdomnen 
 |-  ( A ~< B -> -. A ~~ B )
26 24 25 syl6 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( ran F C. B -> -. A ~~ B ) )
27 9 26 sylbird 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( ran F =/= B -> -. A ~~ B ) )
28 27 necon4ad 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( A ~~ B -> ran F = B ) )
29 5 28 mpd 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F = B )
30 df-fo 
 |-  ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
31 4 29 30 sylanbrc 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> F : A -onto-> B )
32 df-f1o 
 |-  ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
33 1 31 32 sylanbrc 
 |-  ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> F : A -1-1-onto-> B )
34 33 ex 
 |-  ( ( A ~~ B /\ B e. Fin ) -> ( F : A -1-1-> B -> F : A -1-1-onto-> B ) )
35 f1of1 
 |-  ( F : A -1-1-onto-> B -> F : A -1-1-> B )
36 34 35 impbid1 
 |-  ( ( A ~~ B /\ B e. Fin ) -> ( F : A -1-1-> B <-> F : A -1-1-onto-> B ) )