Theorem ffoss 5532

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 5258	. . . 4 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		dffn4 5482	. . . . 5 |- ( F Fn A <-> F : A -onto-> ran F )
3	2	anbi1i 458	. . . 4 |- ( ( F Fn A /\ ran F C_ B ) <-> ( F : A -onto-> ran F /\ ran F C_ B ) )
4	1, 3	bitri 184	. . 3 |- ( F : A --> B <-> ( F : A -onto-> ran F /\ ran F C_ B ) )
5		f11o.1	. . . . 5 |- F e. _V
6	5	rnex 4929	. . . 4 |- ran F e. _V
7		foeq3 5474	. . . . 5 |- ( x = ran F -> ( F : A -onto-> x <-> F : A -onto-> ran F ) )
8		sseq1 3202	. . . . 5 |- ( x = ran F -> ( x C_ B <-> ran F C_ B ) )
9	7, 8	anbi12d 473	. . . 4 |- ( x = ran F -> ( ( F : A -onto-> x /\ x C_ B ) <-> ( F : A -onto-> ran F /\ ran F C_ B ) ) )
10	6, 9	spcev 2855	. . 3 |- ( ( F : A -onto-> ran F /\ ran F C_ B ) -> E. x ( F : A -onto-> x /\ x C_ B ) )
11	4, 10	sylbi 121	. 2 |- ( F : A --> B -> E. x ( F : A -onto-> x /\ x C_ B ) )
12		fof 5476	. . . 4 |- ( F : A -onto-> x -> F : A --> x )
13		fss 5415	. . . 4 |- ( ( F : A --> x /\ x C_ B ) -> F : A --> B )
14	12, 13	sylan 283	. . 3 |- ( ( F : A -onto-> x /\ x C_ B ) -> F : A --> B )
15	14	exlimiv 1609	. 2 |- ( E. x ( F : A -onto-> x /\ x C_ B ) -> F : A --> B )
16	11, 15	impbii 126	1 |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   C_ wss 3153   ran crn 4660   Fn wfn 5249   -->wf 5250   -onto->wfo 5252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-cnv 4667  df-dm 4669  df-rn 4670  df-f 5258  df-fo 5260

This theorem is referenced by:  f11o  5533