Theorem f11o 5334

Description: Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)

Hypothesis

Ref	Expression
f11o.1	|- F e. _V

Assertion

Ref	Expression
f11o	|- ( F : A -1-1-> B <-> E. x ( F : A -1-1-onto-> x /\ x C_ B ) )

Distinct variable groups:   x, F   x, A   x, B

Proof of Theorem f11o

Step	Hyp	Ref	Expression
1		f11o.1	. . . 4 |- F e. _V
2	1	ffoss 5333	. . 3 |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) )
3	2	anbi1i 449	. 2 |- ( ( F : A --> B /\ Fun `' F ) <-> ( E. x ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
4		df-f1 5064	. 2 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5		dff1o3 5307	. . . . . 6 |- ( F : A -1-1-onto-> x <-> ( F : A -onto-> x /\ Fun `' F ) )
6	5	anbi1i 449	. . . . 5 |- ( ( F : A -1-1-onto-> x /\ x C_ B ) <-> ( ( F : A -onto-> x /\ Fun `' F ) /\ x C_ B ) )
7		an32 532	. . . . 5 |- ( ( ( F : A -onto-> x /\ Fun `' F ) /\ x C_ B ) <-> ( ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
8	6, 7	bitri 183	. . . 4 |- ( ( F : A -1-1-onto-> x /\ x C_ B ) <-> ( ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
9	8	exbii 1552	. . 3 |- ( E. x ( F : A -1-1-onto-> x /\ x C_ B ) <-> E. x ( ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
10		19.41v 1841	. . 3 |- ( E. x ( ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) <-> ( E. x ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
11	9, 10	bitri 183	. 2 |- ( E. x ( F : A -1-1-onto-> x /\ x C_ B ) <-> ( E. x ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
12	3, 4, 11	3bitr4i 211	1 |- ( F : A -1-1-> B <-> E. x ( F : A -1-1-onto-> x /\ x C_ B ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1436   e. wcel 1448   _Vcvv 2641   C_ wss 3021   `'ccnv 4476   Fun wfun 5053   -->wf 5055   -1-1->wf1 5056   -onto->wfo 5057   -1-1-onto->wf1o 5058

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-cnv 4485  df-dm 4487  df-rn 4488  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066

This theorem is referenced by:  domen  6575