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Theorem iunxpconst 4999
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 4998 . 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 4962 . 2  |-  ( U_ x  e.  A  {
x }  X.  B
)  =  ( A  X.  B )
41, 3eqtr3i 2433 1  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   {csn 3971   U_ciun 4270    X. cxp 4940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-iun 4272  df-opab 4453  df-xp 4948
This theorem is referenced by:  ralxp  5086  rexxp  5087  mpt2mpt  6331  mpt2mpts  6802  fmpt2  6805  fsumxp  13645  fprodxp  13845  dvfval  22485  indval2  28342  filnetlem3  30596  xpiun  38063
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