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Theorem dff2 5884
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 5594 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fssxp 5605 . . 3  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
31, 2jca 520 . 2  |-  ( F : A --> B  -> 
( F  Fn  A  /\  F  C_  ( A  X.  B ) ) )
4 rnss 5101 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  ran  ( A  X.  B ) )
5 rnxpss 5304 . . . . 5  |-  ran  ( A  X.  B )  C_  B
64, 5syl6ss 3362 . . . 4  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  B )
76anim2i 554 . . 3  |-  ( ( F  Fn  A  /\  F  C_  ( A  X.  B ) )  -> 
( F  Fn  A  /\  ran  F  C_  B
) )
8 df-f 5461 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
97, 8sylibr 205 . 2  |-  ( ( F  Fn  A  /\  F  C_  ( A  X.  B ) )  ->  F : A --> B )
103, 9impbii 182 1  |-  ( F : A --> B  <->  ( F  Fn  A  /\  F  C_  ( A  X.  B
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    C_ wss 3322    X. cxp 4879   ran crn 4882    Fn wfn 5452   -->wf 5453
This theorem is referenced by:  mapval2  7046  cardf2  7835  imasaddflem  13760  imasvscaf  13769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-fun 5459  df-fn 5460  df-f 5461
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