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Theorem constmap 26199
Description: A constant (represented without dummy variables) is an element of a function set.

_Note: In the following development, we will be quite often quantifying over functions and points in N-dimensional space (which are equivalent to functions from an "index set"). Many of the following theorems exist to transfer standard facts about functions to elements of function sets._ (Contributed by Stefan O'Rear, 30-Aug-2014.) (Revised by Stefan O'Rear, 5-May-2015.)

Hypotheses
Ref Expression
constmap.1  |-  A  e. 
_V
constmap.3  |-  C  e. 
_V
Assertion
Ref Expression
constmap  |-  ( B  e.  C  ->  ( A  X.  { B }
)  e.  ( C  ^m  A ) )

Proof of Theorem constmap
StepHypRef Expression
1 fconst6g 5396 . 2  |-  ( B  e.  C  ->  ( A  X.  { B }
) : A --> C )
2 constmap.3 . . 3  |-  C  e. 
_V
3 constmap.1 . . 3  |-  A  e. 
_V
42, 3elmap 6792 . 2  |-  ( ( A  X.  { B } )  e.  ( C  ^m  A )  <-> 
( A  X.  { B } ) : A --> C )
51, 4sylibr 203 1  |-  ( B  e.  C  ->  ( A  X.  { B }
)  e.  ( C  ^m  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1685   _Vcvv 2789   {csn 3641    X. cxp 4686   -->wf 5217  (class class class)co 5820    ^m cmap 6768
This theorem is referenced by:  mzpclall  26216  mzpindd  26235
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-map 6770
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