Theorem f1finf1o 6924

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 525	. . . . 5 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> B e. Fin )
3		f1rn 5404	. . . . . 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 5405	. . . . . . . . 9 |- ( F : A -1-1-> B -> F Fn A )
6		fnima 5316	. . . . . . . . 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 3167	. . . . . . . . 9 |- A C_ A
10	9	a1i 9	. . . . . . . 8 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> A C_ A )
11		simpll 524	. . . . . . . . 9 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> A ~~ B )
12		enfii 6852	. . . . . . . . 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 6770	. . . . . . . 8 |- ( ( F : A -1-1-> B /\ A C_ A /\ A e. Fin ) -> ( F " A ) ~~ A )
15	1, 10, 13, 14	syl3anc 1233	. . . . . . 7 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ( F " A ) ~~ A )
16	8, 15	eqbrtrrd 4013	. . . . . 6 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F ~~ A )
17		entr 6762	. . . . . 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 6909	. . . . 5 |- ( ( B e. Fin /\ ran F C_ B /\ ran F ~~ B ) -> ran F = B )
20	2, 4, 18, 19	syl3anc 1233	. . . 4 |- ( ( ( A ~~ B /\ B e. Fin ) /\ F : A -1-1-> B ) -> ran F = B )
21		dff1o5 5451	. . . 4 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) )
22	1, 20, 21	sylanbrc 415	. . 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 5441	. 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 1348   e. wcel 2141   C_ wss 3121   class class class wbr 3989   ran crn 4612   "cima 4614   Fn wfn 5193   -1-1->wf1 5195   -1-1-onto->wf1o 5197   ~~ cen 6716   Fincfn 6718

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1o 6395  df-er 6513  df-en 6719  df-fin 6721

This theorem is referenced by:  iseqf1olemqf1o  10449  crth  12178  eulerthlemh  12185  pwf1oexmid  14032