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Theorem constmap 26451
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 5565 . 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 6971 . 2  |-  ( ( A  X.  { B } )  e.  ( C  ^m  A )  <-> 
( A  X.  { B } ) : A --> C )
51, 4sylibr 204 1  |-  ( B  e.  C  ->  ( A  X.  { B }
)  e.  ( C  ^m  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   _Vcvv 2892   {csn 3750    X. cxp 4809   -->wf 5383  (class class class)co 6013    ^m cmap 6947
This theorem is referenced by:  mzpclall  26468  mzpindd  26487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-map 6949
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