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Theorem dff2 5443
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 5161 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fssxp 5178 . . 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 4665 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  ran  ( A  X.  B ) )
5 rnxpss 4862 . . . . 5  |-  ran  ( A  X.  B )  C_  B
64, 5syl6ss 3037 . . . 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 5019 . . 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 2999    X. cxp 4436   ran crn 4439    Fn wfn 5010   -->wf 5011
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-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
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-ral 2364  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-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-fun 5017  df-fn 5018  df-f 5019
This theorem is referenced by:  mapval2  6433  frecuzrdgtclt  9824
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