Theorem ffoss 5675

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 5384	. . . 4 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		dffn4 5624	. . . . 5 |- ( F Fn A <-> F : A -onto-> ran F )
3	2	anbi1i 462	. . . 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 5052	. . . 4 |- ran F e. _V
7		foeq3 5616	. . . . 5 |- ( x = ran F -> ( F : A -onto-> x <-> F : A -onto-> ran F ) )
8		sseq1 3271	. . . . 5 |- ( x = ran F -> ( x C_ B <-> ran F C_ B ) )
9	7, 8	anbi12d 477	. . . 4 |- ( x = ran F -> ( ( F : A -onto-> x /\ x C_ B ) <-> ( F : A -onto-> ran F /\ ran F C_ B ) ) )
10	6, 9	spcev 2920	. . 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 5618	. . . 4 |- ( F : A -onto-> x -> F : A --> x )
13		fss 5549	. . . 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 1651	. 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 ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   C_ wss 3220   ran crn 4777   Fn wfn 5375   -->wf 5376   -onto->wfo 5378

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-cnv 4784  df-dm 4786  df-rn 4787  df-f 5384  df-fo 5386

This theorem is used by:  f11o  5676