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Theorem ffoss 5698
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
StepHypRef Expression
1 df-f 5449 . . . 4  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 dffn4 5650 . . . . 5  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
32anbi1i 677 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  <->  ( F : A -onto-> ran  F  /\  ran  F  C_  B ) )
41, 3bitri 241 . . 3  |-  ( F : A --> B  <->  ( F : A -onto-> ran  F  /\  ran  F 
C_  B ) )
5 f11o.1 . . . . 5  |-  F  e. 
_V
65rnex 5124 . . . 4  |-  ran  F  e.  _V
7 foeq3 5642 . . . . 5  |-  ( x  =  ran  F  -> 
( F : A -onto->
x  <->  F : A -onto-> ran  F ) )
8 sseq1 3361 . . . . 5  |-  ( x  =  ran  F  -> 
( x  C_  B  <->  ran 
F  C_  B )
)
97, 8anbi12d 692 . . . 4  |-  ( x  =  ran  F  -> 
( ( F : A -onto-> x  /\  x  C_  B )  <->  ( F : A -onto-> ran  F  /\  ran  F 
C_  B ) ) )
106, 9spcev 3035 . . 3  |-  ( ( F : A -onto-> ran  F  /\  ran  F  C_  B )  ->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
114, 10sylbi 188 . 2  |-  ( F : A --> B  ->  E. x ( F : A -onto-> x  /\  x  C_  B ) )
12 fof 5644 . . . 4  |-  ( F : A -onto-> x  ->  F : A --> x )
13 fss 5590 . . . 4  |-  ( ( F : A --> x  /\  x  C_  B )  ->  F : A --> B )
1412, 13sylan 458 . . 3  |-  ( ( F : A -onto-> x  /\  x  C_  B )  ->  F : A --> B )
1514exlimiv 1644 . 2  |-  ( E. x ( F : A -onto-> x  /\  x  C_  B )  ->  F : A --> B )
1611, 15impbii 181 1  |-  ( F : A --> B  <->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   ran crn 4870    Fn wfn 5440   -->wf 5441   -onto->wfo 5443
This theorem is referenced by:  f11o  5699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-cnv 4877  df-dm 4879  df-rn 4880  df-f 5449  df-fo 5451
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