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Theorem ffoss 5408
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 4650 . . . 4  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 dffn4 5360 . . . . 5  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
32anbi1i 679 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  <->  ( F : A -onto-> ran  F  /\  ran  F  C_  B ) )
41, 3bitri 242 . . 3  |-  ( F : A --> B  <->  ( F : A -onto-> ran  F  /\  ran  F 
C_  B ) )
5 f11o.1 . . . . 5  |-  F  e. 
_V
65rnex 4895 . . . 4  |-  ran  F  e.  _V
7 foeq3 5352 . . . . 5  |-  ( x  =  ran  F  -> 
( F : A -onto->
x  <->  F : A -onto-> ran  F ) )
8 sseq1 3141 . . . . 5  |-  ( x  =  ran  F  -> 
( x  C_  B  <->  ran 
F  C_  B )
)
97, 8anbi12d 694 . . . 4  |-  ( x  =  ran  F  -> 
( ( F : A -onto-> x  /\  x  C_  B )  <->  ( F : A -onto-> ran  F  /\  ran  F 
C_  B ) ) )
106, 9cla4ev 2826 . . 3  |-  ( ( F : A -onto-> ran  F  /\  ran  F  C_  B )  ->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
114, 10sylbi 189 . 2  |-  ( F : A --> B  ->  E. x ( F : A -onto-> x  /\  x  C_  B ) )
12 fof 5354 . . . 4  |-  ( F : A -onto-> x  ->  F : A --> x )
13 fss 5300 . . . 4  |-  ( ( F : A --> x  /\  x  C_  B )  ->  F : A --> B )
1412, 13sylan 459 . . 3  |-  ( ( F : A -onto-> x  /\  x  C_  B )  ->  F : A --> B )
1514exlimiv 2024 . 2  |-  ( E. x ( F : A -onto-> x  /\  x  C_  B )  ->  F : A --> B )
1611, 15impbii 182 1  |-  ( F : A --> B  <->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2740    C_ wss 3094   ran crn 4627    Fn wfn 4633   -->wf 4634   -onto->wfo 4636
This theorem is referenced by:  f11o  5409
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-cnv 4642  df-dm 4644  df-rn 4645  df-f 4650  df-fo 4652
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