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Theorem ixpconstg 7475
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 7427 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  ^m  A
)  =  { f  |  f : A --> B } )
2 vex 3116 . . . . 5  |-  f  e. 
_V
32elixpconst 7474 . . . 4  |-  ( f  e.  X_ x  e.  A  B 
<->  f : A --> B )
43abbi2i 2600 . . 3  |-  X_ x  e.  A  B  =  { f  |  f : A --> B }
51, 4syl6reqr 2527 . 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 1379    e. wcel 1767   {cab 2452   -->wf 5582  (class class class)co 6282    ^m cmap 7417   X_cixp 7466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-ixp 7467
This theorem is referenced by:  ixpconst  7476  mapsnf1o  7507  prdshom  14715  pwsbas  14735  frlmip  18573  pttoponconst  19830  xkoptsub  19887  xkopt  19888  tmdgsum2  20327  rrxip  21554
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