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Theorem fconst2g 6113
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 6077 . . . . . . 7  |-  ( ( F : A --> { B }  /\  x  e.  A
)  ->  ( F `  x )  =  B )
21adantlr 714 . . . . . 6  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  ( F `  x )  =  B )
3 fvconst2g 6112 . . . . . . 7  |-  ( ( B  e.  C  /\  x  e.  A )  ->  ( ( A  X.  { B } ) `  x )  =  B )
43adantll 713 . . . . . 6  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  (
( A  X.  { B } ) `  x
)  =  B )
52, 4eqtr4d 2511 . . . . 5  |-  ( ( ( F : A --> { B }  /\  B  e.  C )  /\  x  e.  A )  ->  ( F `  x )  =  ( ( A  X.  { B }
) `  x )
)
65ralrimiva 2878 . . . 4  |-  ( ( F : A --> { B }  /\  B  e.  C
)  ->  A. x  e.  A  ( F `  x )  =  ( ( A  X.  { B } ) `  x
) )
7 ffn 5729 . . . . 5  |-  ( F : A --> { B }  ->  F  Fn  A
)
8 fnconstg 5771 . . . . 5  |-  ( B  e.  C  ->  ( A  X.  { B }
)  Fn  A )
9 eqfnfv 5973 . . . . 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 477 . . . 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 435 . 2  |-  ( B  e.  C  ->  ( F : A --> { B }  ->  F  =  ( A  X.  { B } ) ) )
13 fconstg 5770 . . 3  |-  ( B  e.  C  ->  ( A  X.  { B }
) : A --> { B } )
14 feq1 5711 . . 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 369    = wceq 1379    e. wcel 1767   A.wral 2814   {csn 4027    X. cxp 4997    Fn wfn 5581   -->wf 5582   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594
This theorem is referenced by:  fconst2  6115  fconst5  6116  repsdf2  12709  cnconst  19551
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