Theorem ffoss 5178

Description: Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)

Hypothesis

Ref	Expression
f11o.1	|- F e. _V

Assertion

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

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

Proof of Theorem ffoss

Step	Hyp	Ref	Expression
1		df-f 4926	. . . 4 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		dffn4 5132	. . . . 5 |- ( F Fn A <-> F : A -onto-> ran F )
3	2	anbi1i 445	. . . 4 |- ( ( F Fn A /\ ran F C_ B ) <-> ( F : A -onto-> ran F /\ ran F C_ B ) )
4	1, 3	bitri 182	. . 3 |- ( F : A --> B <-> ( F : A -onto-> ran F /\ ran F C_ B ) )
5		f11o.1	. . . . 5 |- F e. _V
6	5	rnex 4617	. . . 4 |- ran F e. _V
7		foeq3 5124	. . . . 5 |- ( x = ran F -> ( F : A -onto-> x <-> F : A -onto-> ran F ) )
8		sseq1 3020	. . . . 5 |- ( x = ran F -> ( x C_ B <-> ran F C_ B ) )
9	7, 8	anbi12d 456	. . . 4 |- ( x = ran F -> ( ( F : A -onto-> x /\ x C_ B ) <-> ( F : A -onto-> ran F /\ ran F C_ B ) ) )
10	6, 9	spcev 2692	. . 3 |- ( ( F : A -onto-> ran F /\ ran F C_ B ) -> E. x ( F : A -onto-> x /\ x C_ B ) )
11	4, 10	sylbi 119	. 2 |- ( F : A --> B -> E. x ( F : A -onto-> x /\ x C_ B ) )
12		fof 5126	. . . 4 |- ( F : A -onto-> x -> F : A --> x )
13		fss 5074	. . . 4 |- ( ( F : A --> x /\ x C_ B ) -> F : A --> B )
14	12, 13	sylan 277	. . 3 |- ( ( F : A -onto-> x /\ x C_ B ) -> F : A --> B )
15	14	exlimiv 1529	. 2 |- ( E. x ( F : A -onto-> x /\ x C_ B ) -> F : A --> B )
16	11, 15	impbii 124	1 |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   C_ wss 2973   ran crn 4364   Fn wfn 4917   -->wf 4918   -onto->wfo 4920

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-cnv 4371  df-dm 4373  df-rn 4374  df-f 4926  df-fo 4928

This theorem is referenced by:  f11o  5179