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Theorem constmap 26294
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 5536 . 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 6939 . 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 1715   _Vcvv 2873   {csn 3729    X. cxp 4790   -->wf 5354  (class class class)co 5981    ^m cmap 6915
This theorem is referenced by:  mzpclall  26311  mzpindd  26330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-map 6917
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