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Theorem fconst2g 6105
Description: A constant function expressed as a Cartesian 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 6068 . . . . . . 7  |-  ( ( F : A --> { B }  /\  x  e.  A
)  ->  ( F `  x )  =  B )
21adantlr 713 . . . . . 6  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  ( F `  x )  =  B )
3 fvconst2g 6104 . . . . . . 7  |-  ( ( B  e.  C  /\  x  e.  A )  ->  ( ( A  X.  { B } ) `  x )  =  B )
43adantll 712 . . . . . 6  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  (
( A  X.  { B } ) `  x
)  =  B )
52, 4eqtr4d 2446 . . . . 5  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  ( F `  x )  =  ( ( A  X.  { B }
) `  x )
)
65ralrimiva 2817 . . . 4  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  A. x  e.  A  ( F `  x )  =  ( ( A  X.  { B } ) `  x
) )
7 ffn 5713 . . . . 5  |-  ( F : A --> { B }  ->  F  Fn  A
)
8 fnconstg 5755 . . . . 5  |-  ( B  e.  C  ->  ( A  X.  { B }
)  Fn  A )
9 eqfnfv 5958 . . . . 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 475 . . . 4  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  ( F  =  ( A  X.  { B } )  <->  A. x  e.  A  ( F `  x )  =  ( ( A  X.  { B } ) `  x
) ) )
116, 10mpbird 232 . . 3  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  F  =  ( A  X.  { B } ) )
1211expcom 433 . 2  |-  ( B  e.  C  ->  ( F : A --> { B }  ->  F  =  ( A  X.  { B } ) ) )
13 fconstg 5754 . . 3  |-  ( B  e.  C  ->  ( A  X.  { B }
) : A --> { B } )
14 feq1 5695 . . 3  |-  ( F  =  ( A  X.  { B } )  -> 
( F : A --> { B }  <->  ( A  X.  { B } ) : A --> { B } ) )
1513, 14syl5ibrcom 222 . 2  |-  ( B  e.  C  ->  ( F  =  ( A  X.  { B } )  ->  F : A --> { B } ) )
1612, 15impbid 191 1  |-  ( B  e.  C  ->  ( F : A --> { B } 
<->  F  =  ( A  X.  { B }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   {csn 3971    X. cxp 4820    Fn wfn 5563   -->wf 5564   ` cfv 5568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576
This theorem is referenced by:  fconst2  6107  fconst5  6108  repsdf2  12804  cnconst  20076  padct  27978
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