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Theorem iunxpconst 4995
Description: Membership in a union of Cartesian 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 4994 . 2  |-  ( U_ x  e.  A  {
x }  X.  B
)  =  U_ x  e.  A  ( {
x }  X.  B
)
2 iunid 4325 . . 3  |-  U_ x  e.  A  { x }  =  A
32xpeq1i 4960 . 2  |-  ( U_ x  e.  A  {
x }  X.  B
)  =  ( A  X.  B )
41, 3eqtr3i 2482 1  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   {csn 3977   U_ciun 4271    X. cxp 4938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-iun 4273  df-opab 4451  df-xp 4946
This theorem is referenced by:  ralxp  5081  rexxp  5082  mpt2mpt  6284  mpt2mpts  6740  fmpt2  6743  fsumxp  13343  dvfval  21490  indval2  26607  fprodxp  27629  filnetlem3  28741
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