Theorem f11o 7128

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 7127	. . 3 |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) )
3	2	anbi1i 731	. 2 |- ( ( F : A --> B /\ Fun `' F ) <-> ( E. x ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
4		df-f1 5893	. 2 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5		dff1o3 6143	. . . . . 6 |- ( F : A -1-1-onto-> x <-> ( F : A -onto-> x /\ Fun `' F ) )
6	5	anbi1i 731	. . . . 5 |- ( ( F : A -1-1-onto-> x /\ x C_ B ) <-> ( ( F : A -onto-> x /\ Fun `' F ) /\ x C_ B ) )
7		an32 839	. . . . 5 |- ( ( ( F : A -onto-> x /\ Fun `' F ) /\ x C_ B ) <-> ( ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
8	6, 7	bitri 264	. . . 4 |- ( ( F : A -1-1-onto-> x /\ x C_ B ) <-> ( ( F : A -onto-> x /\ x C_ B ) /\ Fun `' F ) )
9	8	exbii 1774	. . 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 1914	. . 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 264	. 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 292	1 |- ( F : A -1-1-> B <-> E. x ( F : A -1-1-onto-> x /\ x C_ B ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   _Vcvv 3200   C_ wss 3574   `'ccnv 5113   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  domen  7968  uspgrsprfo  41756