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Theorem ixpconstg 6841
Description: Infinite Cartesian product of a constant  B. (Contributed by Mario Carneiro, 11-Jan-2015.)
Assertion
Ref Expression
ixpconstg  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
X_ x  e.  A  B  =  ( B  ^m  A ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    V( x)    W( x)

Proof of Theorem ixpconstg
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 mapvalg 6798 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  ^m  A
)  =  { f  |  f : A --> B } )
2 vex 2804 . . . . 5  |-  f  e. 
_V
32elixpconst 6840 . . . 4  |-  ( f  e.  X_ x  e.  A  B 
<->  f : A --> B )
43abbi2i 2407 . . 3  |-  X_ x  e.  A  B  =  { f  |  f : A --> B }
51, 4syl6reqr 2347 . 2  |-  ( ( B  e.  W  /\  A  e.  V )  -> 
X_ x  e.  A  B  =  ( B  ^m  A ) )
65ancoms 439 1  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
X_ x  e.  A  B  =  ( B  ^m  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   -->wf 5267  (class class class)co 5874    ^m cmap 6788   X_cixp 6833
This theorem is referenced by:  ixpconst  6842  mapsnf1o  6873  prdshom  13382  pwsbas  13402  pttoponconst  17308  xkoptsub  17364  xkopt  17365  tmdgsum2  17795  istopxc  25651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-ixp 6834
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