Theorem ffoss 5392

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 5122	. . . 4 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		dffn4 5346	. . . . 5 |- ( F Fn A <-> F : A -onto-> ran F )
3	2	anbi1i 453	. . . 4 |- ( ( F Fn A /\ ran F C_ B ) <-> ( F : A -onto-> ran F /\ ran F C_ B ) )
4	1, 3	bitri 183	. . 3 |- ( F : A --> B <-> ( F : A -onto-> ran F /\ ran F C_ B ) )
5		f11o.1	. . . . 5 |- F e. _V
6	5	rnex 4801	. . . 4 |- ran F e. _V
7		foeq3 5338	. . . . 5 |- ( x = ran F -> ( F : A -onto-> x <-> F : A -onto-> ran F ) )
8		sseq1 3115	. . . . 5 |- ( x = ran F -> ( x C_ B <-> ran F C_ B ) )
9	7, 8	anbi12d 464	. . . 4 |- ( x = ran F -> ( ( F : A -onto-> x /\ x C_ B ) <-> ( F : A -onto-> ran F /\ ran F C_ B ) ) )
10	6, 9	spcev 2775	. . 3 |- ( ( F : A -onto-> ran F /\ ran F C_ B ) -> E. x ( F : A -onto-> x /\ x C_ B ) )
11	4, 10	sylbi 120	. 2 |- ( F : A --> B -> E. x ( F : A -onto-> x /\ x C_ B ) )
12		fof 5340	. . . 4 |- ( F : A -onto-> x -> F : A --> x )
13		fss 5279	. . . 4 |- ( ( F : A --> x /\ x C_ B ) -> F : A --> B )
14	12, 13	sylan 281	. . 3 |- ( ( F : A -onto-> x /\ x C_ B ) -> F : A --> B )
15	14	exlimiv 1577	. 2 |- ( E. x ( F : A -onto-> x /\ x C_ B ) -> F : A --> B )
16	11, 15	impbii 125	1 |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   _Vcvv 2681   C_ wss 3066   ran crn 4535   Fn wfn 5113   -->wf 5114   -onto->wfo 5116

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-fo 5124

This theorem is referenced by:  f11o  5393