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Theorem fconst2g 5726
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 5699 . . . . . . 7  |-  ( ( F : A --> { B }  /\  x  e.  A
)  ->  ( F `  x )  =  B )
21adantlr 477 . . . . . 6  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  ( F `  x )  =  B )
3 fvconst2g 5725 . . . . . . 7  |-  ( ( B  e.  C  /\  x  e.  A )  ->  ( ( A  X.  { B } ) `  x )  =  B )
43adantll 476 . . . . . 6  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  (
( A  X.  { B } ) `  x
)  =  B )
52, 4eqtr4d 2213 . . . . 5  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  ( F `  x )  =  ( ( A  X.  { B }
) `  x )
)
65ralrimiva 2550 . . . 4  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  A. x  e.  A  ( F `  x )  =  ( ( A  X.  { B } ) `  x
) )
7 ffn 5360 . . . . 5  |-  ( F : A --> { B }  ->  F  Fn  A
)
8 fnconstg 5408 . . . . 5  |-  ( B  e.  C  ->  ( A  X.  { B }
)  Fn  A )
9 eqfnfv 5608 . . . . 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 289 . . . 4  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  ( F  =  ( A  X.  { B } )  <->  A. x  e.  A  ( F `  x )  =  ( ( A  X.  { B } ) `  x
) ) )
116, 10mpbird 167 . . 3  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  F  =  ( A  X.  { B } ) )
1211expcom 116 . 2  |-  ( B  e.  C  ->  ( F : A --> { B }  ->  F  =  ( A  X.  { B } ) ) )
13 fconstg 5407 . . 3  |-  ( B  e.  C  ->  ( A  X.  { B }
) : A --> { B } )
14 feq1 5343 . . 3  |-  ( F  =  ( A  X.  { B } )  -> 
( F : A --> { B }  <->  ( A  X.  { B } ) : A --> { B } ) )
1513, 14syl5ibrcom 157 . 2  |-  ( B  e.  C  ->  ( F  =  ( A  X.  { B } )  ->  F : A --> { B } ) )
1612, 15impbid 129 1  |-  ( B  e.  C  ->  ( F : A --> { B } 
<->  F  =  ( A  X.  { B }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   {csn 3591    X. cxp 4620    Fn wfn 5206   -->wf 5207   ` cfv 5211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219
This theorem is referenced by:  fconst2  5728  cnconst  13367  nninfall  14381
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