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Theorem dff2 5945
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 5639 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fssxp 5651 . . 3  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
31, 2jca 530 . 2  |-  ( F : A --> B  -> 
( F  Fn  A  /\  F  C_  ( A  X.  B ) ) )
4 rnss 5144 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  ran  ( A  X.  B ) )
5 rnxpss 5349 . . . . 5  |-  ran  ( A  X.  B )  C_  B
64, 5syl6ss 3429 . . . 4  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  B )
76anim2i 567 . . 3  |-  ( ( F  Fn  A  /\  F  C_  ( A  X.  B ) )  -> 
( F  Fn  A  /\  ran  F  C_  B
) )
8 df-f 5500 . . 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 367    C_ wss 3389    X. cxp 4911   ran crn 4914    Fn wfn 5491   -->wf 5492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-dm 4923  df-rn 4924  df-fun 5498  df-fn 5499  df-f 5500
This theorem is referenced by:  mapval2  7367  cardf2  8237  imasaddflem  14937  imasvscaf  14946
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