Theorem ffoss 5285

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 5019	. . . 4 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		dffn4 5239	. . . . 5 |- ( F Fn A <-> F : A -onto-> ran F )
3	2	anbi1i 446	. . . 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 4700	. . . 4 |- ran F e. _V
7		foeq3 5231	. . . . 5 |- ( x = ran F -> ( F : A -onto-> x <-> F : A -onto-> ran F ) )
8		sseq1 3047	. . . . 5 |- ( x = ran F -> ( x C_ B <-> ran F C_ B ) )
9	7, 8	anbi12d 457	. . . 4 |- ( x = ran F -> ( ( F : A -onto-> x /\ x C_ B ) <-> ( F : A -onto-> ran F /\ ran F C_ B ) ) )
10	6, 9	spcev 2713	. . 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 5233	. . . 4 |- ( F : A -onto-> x -> F : A --> x )
13		fss 5172	. . . 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 1534	. 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 1289   E.wex 1426   e. wcel 1438   _Vcvv 2619   C_ wss 2999   ran crn 4439   Fn wfn 5010   -->wf 5011   -onto->wfo 5013

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-cnv 4446  df-dm 4448  df-rn 4449  df-f 5019  df-fo 5021

This theorem is referenced by:  f11o  5286