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Theorem ixpconstg 6725
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 vex 2755 . . . . 5  |-  f  e. 
_V
21elixpconst 6724 . . . 4  |-  ( f  e.  X_ x  e.  A  B 
<->  f : A --> B )
32abbi2i 2304 . . 3  |-  X_ x  e.  A  B  =  { f  |  f : A --> B }
4 mapvalg 6676 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  ^m  A
)  =  { f  |  f : A --> B } )
53, 4eqtr4id 2241 . 2  |-  ( ( B  e.  W  /\  A  e.  V )  -> 
X_ x  e.  A  B  =  ( B  ^m  A ) )
65ancoms 268 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 104    = wceq 1364    e. wcel 2160   {cab 2175   -->wf 5227  (class class class)co 5891    ^m cmap 6666   X_cixp 6716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-map 6668  df-ixp 6717
This theorem is referenced by:  ixpconst  6726  mapsnf1o  6755
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