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Theorem constmap 26156
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 5368 . 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 6764 . 2  |-  ( ( A  X.  { B } )  e.  ( C  ^m  A )  <-> 
( A  X.  { B } ) : A --> C )
51, 4sylibr 205 1  |-  ( B  e.  C  ->  ( A  X.  { B }
)  e.  ( C  ^m  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621   _Vcvv 2763   {csn 3614    X. cxp 4659   -->wf 4669  (class class class)co 5792    ^m cmap 6740
This theorem is referenced by:  mzpclall  26173  mzpindd  26192
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-map 6742
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