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Theorem fconst2g 5635
Description: A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)
Assertion
Ref Expression
fconst2g  |-  ( B  e.  C  ->  ( F : A --> { B } 
<->  F  =  ( A  X.  { B }
) ) )

Proof of Theorem fconst2g
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fvconst 5608 . . . . . . 7  |-  ( ( F : A --> { B }  /\  x  e.  A
)  ->  ( F `  x )  =  B )
21adantlr 468 . . . . . 6  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  ( F `  x )  =  B )
3 fvconst2g 5634 . . . . . . 7  |-  ( ( B  e.  C  /\  x  e.  A )  ->  ( ( A  X.  { B } ) `  x )  =  B )
43adantll 467 . . . . . 6  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  (
( A  X.  { B } ) `  x
)  =  B )
52, 4eqtr4d 2175 . . . . 5  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  ( F `  x )  =  ( ( A  X.  { B }
) `  x )
)
65ralrimiva 2505 . . . 4  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  A. x  e.  A  ( F `  x )  =  ( ( A  X.  { B } ) `  x
) )
7 ffn 5272 . . . . 5  |-  ( F : A --> { B }  ->  F  Fn  A
)
8 fnconstg 5320 . . . . 5  |-  ( B  e.  C  ->  ( A  X.  { B }
)  Fn  A )
9 eqfnfv 5518 . . . . 5  |-  ( ( F  Fn  A  /\  ( A  X.  { B } )  Fn  A
)  ->  ( F  =  ( A  X.  { B } )  <->  A. x  e.  A  ( F `  x )  =  ( ( A  X.  { B } ) `  x
) ) )
107, 8, 9syl2an 287 . . . 4  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  ( F  =  ( A  X.  { B } )  <->  A. x  e.  A  ( F `  x )  =  ( ( A  X.  { B } ) `  x
) ) )
116, 10mpbird 166 . . 3  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  F  =  ( A  X.  { B } ) )
1211expcom 115 . 2  |-  ( B  e.  C  ->  ( F : A --> { B }  ->  F  =  ( A  X.  { B } ) ) )
13 fconstg 5319 . . 3  |-  ( B  e.  C  ->  ( A  X.  { B }
) : A --> { B } )
14 feq1 5255 . . 3  |-  ( F  =  ( A  X.  { B } )  -> 
( F : A --> { B }  <->  ( A  X.  { B } ) : A --> { B } ) )
1513, 14syl5ibrcom 156 . 2  |-  ( B  e.  C  ->  ( F  =  ( A  X.  { B } )  ->  F : A --> { B } ) )
1612, 15impbid 128 1  |-  ( B  e.  C  ->  ( F : A --> { B } 
<->  F  =  ( A  X.  { B }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2416   {csn 3527    X. cxp 4537    Fn wfn 5118   -->wf 5119   ` cfv 5123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131
This theorem is referenced by:  fconst2  5637  cnconst  12413  nninfall  13234
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