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Theorem ixpconstg 7497
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 7448 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  ^m  A
)  =  { f  |  f : A --> B } )
2 vex 3112 . . . . 5  |-  f  e. 
_V
32elixpconst 7496 . . . 4  |-  ( f  e.  X_ x  e.  A  B 
<->  f : A --> B )
43abbi2i 2590 . . 3  |-  X_ x  e.  A  B  =  { f  |  f : A --> B }
51, 4syl6reqr 2517 . 2  |-  ( ( B  e.  W  /\  A  e.  V )  -> 
X_ x  e.  A  B  =  ( B  ^m  A ) )
65ancoms 453 1  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
X_ x  e.  A  B  =  ( B  ^m  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   -->wf 5590  (class class class)co 6296    ^m cmap 7438   X_cixp 7488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-ixp 7489
This theorem is referenced by:  ixpconst  7498  mapsnf1o  7529  prdshom  14883  pwsbas  14903  frlmip  18935  pttoponconst  20223  xkoptsub  20280  xkopt  20281  tmdgsum2  20720  rrxip  21947
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