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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
StepHypRef 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 )
32anbi1i 446 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  <->  ( F : A -onto-> ran  F  /\  ran  F  C_  B ) )
41, 3bitri 182 . . 3  |-  ( F : A --> B  <->  ( F : A -onto-> ran  F  /\  ran  F 
C_  B ) )
5 f11o.1 . . . . 5  |-  F  e. 
_V
65rnex 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 )
)
97, 8anbi12d 457 . . . 4  |-  ( x  =  ran  F  -> 
( ( F : A -onto-> x  /\  x  C_  B )  <->  ( F : A -onto-> ran  F  /\  ran  F 
C_  B ) ) )
106, 9spcev 2713 . . 3  |-  ( ( F : A -onto-> ran  F  /\  ran  F  C_  B )  ->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
114, 10sylbi 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 )
1412, 13sylan 277 . . 3  |-  ( ( F : A -onto-> x  /\  x  C_  B )  ->  F : A --> B )
1514exlimiv 1534 . 2  |-  ( E. x ( F : A -onto-> x  /\  x  C_  B )  ->  F : A --> B )
1611, 15impbii 124 1  |-  ( F : A --> B  <->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
Colors of variables: wff set class
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
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