MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fconst2g Unicode version

Theorem fconst2g 5744
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 5724 . . . . . . 7  |-  ( ( F : A --> { B }  /\  x  e.  A
)  ->  ( F `  x )  =  B )
21adantlr 695 . . . . . 6  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  ( F `  x )  =  B )
3 fvconst2g 5743 . . . . . . 7  |-  ( ( B  e.  C  /\  x  e.  A )  ->  ( ( A  X.  { B } ) `  x )  =  B )
43adantll 694 . . . . . 6  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  (
( A  X.  { B } ) `  x
)  =  B )
52, 4eqtr4d 2331 . . . . 5  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  ( F `  x )  =  ( ( A  X.  { B }
) `  x )
)
65ralrimiva 2639 . . . 4  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  A. x  e.  A  ( F `  x )  =  ( ( A  X.  { B } ) `  x
) )
7 ffn 5405 . . . . 5  |-  ( F : A --> { B }  ->  F  Fn  A
)
8 fnconstg 5445 . . . . 5  |-  ( B  e.  C  ->  ( A  X.  { B }
)  Fn  A )
9 eqfnfv 5638 . . . . 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 463 . . . 4  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  ( F  =  ( A  X.  { B } )  <->  A. x  e.  A  ( F `  x )  =  ( ( A  X.  { B } ) `  x
) ) )
116, 10mpbird 223 . . 3  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  F  =  ( A  X.  { B } ) )
1211expcom 424 . 2  |-  ( B  e.  C  ->  ( F : A --> { B }  ->  F  =  ( A  X.  { B } ) ) )
13 fconstg 5444 . . 3  |-  ( B  e.  C  ->  ( A  X.  { B }
) : A --> { B } )
14 feq1 5391 . . 3  |-  ( F  =  ( A  X.  { B } )  -> 
( F : A --> { B }  <->  ( A  X.  { B } ) : A --> { B } ) )
1513, 14syl5ibrcom 213 . 2  |-  ( B  e.  C  ->  ( F  =  ( A  X.  { B } )  ->  F : A --> { B } ) )
1612, 15impbid 183 1  |-  ( B  e.  C  ->  ( F : A --> { B } 
<->  F  =  ( A  X.  { B }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {csn 3653    X. cxp 4703    Fn wfn 5266   -->wf 5267   ` cfv 5271
This theorem is referenced by:  fconst2  5746  fconst5  5747  cnconst  17028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279
  Copyright terms: Public domain W3C validator