Theorem f11o 5393

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 5392	. . 3 |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) )
3	2	anbi1i 453	. 2 |- ( ( F : A --> B /\ Fun `' F ) <-> ( E. x ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
4		df-f1 5123	. 2 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5		dff1o3 5366	. . . . . 6 |- ( F : A -1-1-onto-> x <-> ( F : A -onto-> x /\ Fun `' F ) )
6	5	anbi1i 453	. . . . 5 |- ( ( F : A -1-1-onto-> x /\ x C_ B ) <-> ( ( F : A -onto-> x /\ Fun `' F ) /\ x C_ B ) )
7		an32 551	. . . . 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 1584	. . 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 1874	. . 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 1468   e. wcel 1480   _Vcvv 2681   C_ wss 3066   `'ccnv 4533   Fun wfun 5112   -->wf 5114   -1-1->wf1 5115   -onto->wfo 5116   -1-1-onto->wf1o 5117

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-cnv 4542  df-dm 4544  df-rn 4545  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125

This theorem is referenced by:  domen  6638