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Theorem dff2 5959
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 5662 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fssxp 5673 . . 3  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
31, 2jca 532 . 2  |-  ( F : A --> B  -> 
( F  Fn  A  /\  F  C_  ( A  X.  B ) ) )
4 rnss 5171 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  ran  ( A  X.  B ) )
5 rnxpss 5373 . . . . 5  |-  ran  ( A  X.  B )  C_  B
64, 5syl6ss 3471 . . . 4  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  B )
76anim2i 569 . . 3  |-  ( ( F  Fn  A  /\  F  C_  ( A  X.  B ) )  -> 
( F  Fn  A  /\  ran  F  C_  B
) )
8 df-f 5525 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
97, 8sylibr 212 . 2  |-  ( ( F  Fn  A  /\  F  C_  ( A  X.  B ) )  ->  F : A --> B )
103, 9impbii 188 1  |-  ( F : A --> B  <->  ( F  Fn  A  /\  F  C_  ( A  X.  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    C_ wss 3431    X. cxp 4941   ran crn 4944    Fn wfn 5516   -->wf 5517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-xp 4949  df-rel 4950  df-cnv 4951  df-dm 4953  df-rn 4954  df-fun 5523  df-fn 5524  df-f 5525
This theorem is referenced by:  mapval2  7347  cardf2  8219  imasaddflem  14582  imasvscaf  14591
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