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Theorem iunxpconst 4746
Description: Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
Assertion
Ref Expression
iunxpconst  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem iunxpconst
StepHypRef Expression
1 xpiundir 4744 . 2  |-  ( U_ x  e.  A  {
x }  X.  B
)  =  U_ x  e.  A  ( {
x }  X.  B
)
2 iunid 3957 . . 3  |-  U_ x  e.  A  { x }  =  A
32xpeq1i 4709 . 2  |-  ( U_ x  e.  A  {
x }  X.  B
)  =  ( A  X.  B )
41, 3eqtr3i 2305 1  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   {csn 3640   U_ciun 3905    X. cxp 4687
This theorem is referenced by:  ralxp  4827  rexxp  4828  mpt2mpt  5939  mpt2mpts  6188  fmpt2  6191  fsumxp  12235  dvfval  19247  indval2  23598  filnetlem3  26329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iun 3907  df-opab 4078  df-xp 4695
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