Theorem f11o 5605

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 5604	. . 3 |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) )
3	2	anbi1i 458	. 2 |- ( ( F : A --> B /\ Fun `' F ) <-> ( E. x ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
4		df-f1 5323	. 2 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5		dff1o3 5578	. . . . . 6 |- ( F : A -1-1-onto-> x <-> ( F : A -onto-> x /\ Fun `' F ) )
6	5	anbi1i 458	. . . . 5 |- ( ( F : A -1-1-onto-> x /\ x C_ B ) <-> ( ( F : A -onto-> x /\ Fun `' F ) /\ x C_ B ) )
7		an32 562	. . . . 5 |- ( ( ( F : A -onto-> x /\ Fun `' F ) /\ x C_ B ) <-> ( ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
8	6, 7	bitri 184	. . . 4 |- ( ( F : A -1-1-onto-> x /\ x C_ B ) <-> ( ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
9	8	exbii 1651	. . 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 1949	. . 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 184	. 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 212	1 |- ( F : A -1-1-> B <-> E. x ( F : A -1-1-onto-> x /\ x C_ B ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1538   e. wcel 2200   _Vcvv 2799   C_ wss 3197   `'ccnv 4718   Fun wfun 5312   -->wf 5314   -1-1->wf1 5315   -onto->wfo 5316   -1-1-onto->wf1o 5317

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-cnv 4727  df-dm 4729  df-rn 4730  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325

This theorem is referenced by:  domen  6900