Theorem f11o 5465

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 5464	. . 3 |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) )
3	2	anbi1i 454	. 2 |- ( ( F : A --> B /\ Fun `' F ) <-> ( E. x ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
4		df-f1 5193	. 2 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5		dff1o3 5438	. . . . . 6 |- ( F : A -1-1-onto-> x <-> ( F : A -onto-> x /\ Fun `' F ) )
6	5	anbi1i 454	. . . . 5 |- ( ( F : A -1-1-onto-> x /\ x C_ B ) <-> ( ( F : A -onto-> x /\ Fun `' F ) /\ x C_ B ) )
7		an32 552	. . . . 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 1593	. . 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 1890	. . 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 1480   e. wcel 2136   _Vcvv 2726   C_ wss 3116   `'ccnv 4603   Fun wfun 5182   -->wf 5184   -1-1->wf1 5185   -onto->wfo 5186   -1-1-onto->wf1o 5187

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-cnv 4612  df-dm 4614  df-rn 4615  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195

This theorem is referenced by:  domen  6717