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Theorem ffoss 5509
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 5236 . . . 4  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 dffn4 5460 . . . . 5  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
32anbi1i 458 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  <->  ( F : A -onto-> ran  F  /\  ran  F  C_  B ) )
41, 3bitri 184 . . 3  |-  ( F : A --> B  <->  ( F : A -onto-> ran  F  /\  ran  F 
C_  B ) )
5 f11o.1 . . . . 5  |-  F  e. 
_V
65rnex 4909 . . . 4  |-  ran  F  e.  _V
7 foeq3 5452 . . . . 5  |-  ( x  =  ran  F  -> 
( F : A -onto->
x  <->  F : A -onto-> ran  F ) )
8 sseq1 3193 . . . . 5  |-  ( x  =  ran  F  -> 
( x  C_  B  <->  ran 
F  C_  B )
)
97, 8anbi12d 473 . . . 4  |-  ( x  =  ran  F  -> 
( ( F : A -onto-> x  /\  x  C_  B )  <->  ( F : A -onto-> ran  F  /\  ran  F 
C_  B ) ) )
106, 9spcev 2847 . . 3  |-  ( ( F : A -onto-> ran  F  /\  ran  F  C_  B )  ->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
114, 10sylbi 121 . 2  |-  ( F : A --> B  ->  E. x ( F : A -onto-> x  /\  x  C_  B ) )
12 fof 5454 . . . 4  |-  ( F : A -onto-> x  ->  F : A --> x )
13 fss 5393 . . . 4  |-  ( ( F : A --> x  /\  x  C_  B )  ->  F : A --> B )
1412, 13sylan 283 . . 3  |-  ( ( F : A -onto-> x  /\  x  C_  B )  ->  F : A --> B )
1514exlimiv 1609 . 2  |-  ( E. x ( F : A -onto-> x  /\  x  C_  B )  ->  F : A --> B )
1611, 15impbii 126 1  |-  ( F : A --> B  <->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160   _Vcvv 2752    C_ wss 3144   ran crn 4642    Fn wfn 5227   -->wf 5228   -onto->wfo 5230
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-cnv 4649  df-dm 4651  df-rn 4652  df-f 5236  df-fo 5238
This theorem is referenced by:  f11o  5510
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