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Theorem ffoss 5505
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 5259 . . . 4  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 dffn4 5457 . . . . 5  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
32anbi1i 676 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  <->  ( F : A -onto-> ran  F  /\  ran  F  C_  B ) )
41, 3bitri 240 . . 3  |-  ( F : A --> B  <->  ( F : A -onto-> ran  F  /\  ran  F 
C_  B ) )
5 f11o.1 . . . . 5  |-  F  e. 
_V
65rnex 4942 . . . 4  |-  ran  F  e.  _V
7 foeq3 5449 . . . . 5  |-  ( x  =  ran  F  -> 
( F : A -onto->
x  <->  F : A -onto-> ran  F ) )
8 sseq1 3199 . . . . 5  |-  ( x  =  ran  F  -> 
( x  C_  B  <->  ran 
F  C_  B )
)
97, 8anbi12d 691 . . . 4  |-  ( x  =  ran  F  -> 
( ( F : A -onto-> x  /\  x  C_  B )  <->  ( F : A -onto-> ran  F  /\  ran  F 
C_  B ) ) )
106, 9spcev 2875 . . 3  |-  ( ( F : A -onto-> ran  F  /\  ran  F  C_  B )  ->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
114, 10sylbi 187 . 2  |-  ( F : A --> B  ->  E. x ( F : A -onto-> x  /\  x  C_  B ) )
12 fof 5451 . . . 4  |-  ( F : A -onto-> x  ->  F : A --> x )
13 fss 5397 . . . 4  |-  ( ( F : A --> x  /\  x  C_  B )  ->  F : A --> B )
1412, 13sylan 457 . . 3  |-  ( ( F : A -onto-> x  /\  x  C_  B )  ->  F : A --> B )
1514exlimiv 1666 . 2  |-  ( E. x ( F : A -onto-> x  /\  x  C_  B )  ->  F : A --> B )
1611, 15impbii 180 1  |-  ( F : A --> B  <->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ran crn 4690    Fn wfn 5250   -->wf 5251   -onto->wfo 5253
This theorem is referenced by:  f11o  5506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700  df-f 5259  df-fo 5261
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