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Theorem dff2 5363
Description: Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
Assertion
Ref Expression
dff2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  F  C_  ( A  X.  B
) ) )

Proof of Theorem dff2
StepHypRef Expression
1 ffn 5097 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fssxp 5109 . . 3  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
31, 2jca 300 . 2  |-  ( F : A --> B  -> 
( F  Fn  A  /\  F  C_  ( A  X.  B ) ) )
4 rnss 4612 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  ran  ( A  X.  B ) )
5 rnxpss 4804 . . . . 5  |-  ran  ( A  X.  B )  C_  B
64, 5syl6ss 3020 . . . 4  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  B )
76anim2i 334 . . 3  |-  ( ( F  Fn  A  /\  F  C_  ( A  X.  B ) )  -> 
( F  Fn  A  /\  ran  F  C_  B
) )
8 df-f 4956 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
97, 8sylibr 132 . 2  |-  ( ( F  Fn  A  /\  F  C_  ( A  X.  B ) )  ->  F : A --> B )
103, 9impbii 124 1  |-  ( F : A --> B  <->  ( F  Fn  A  /\  F  C_  ( A  X.  B
) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    C_ wss 2982    X. cxp 4389   ran crn 4392    Fn wfn 4947   -->wf 4948
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402  df-fun 4954  df-fn 4955  df-f 4956
This theorem is referenced by:  frecuzrdgtclt  9555
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