Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  constmap Unicode version

Theorem constmap 26799
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 5432 . 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 6798 . 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 1686   _Vcvv 2790   {csn 3642    X. cxp 4689   -->wf 5253  (class class class)co 5860    ^m cmap 6774
This theorem is referenced by:  mzpclall  26816  mzpindd  26835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-map 6776
  Copyright terms: Public domain W3C validator