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Theorem iunxpconst 4671
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 4670 . 2  |-  ( U_ x  e.  A  {
x }  X.  B
)  =  U_ x  e.  A  ( {
x }  X.  B
)
2 iunid 3928 . . 3  |-  U_ x  e.  A  { x }  =  A
32xpeq1i 4631 . 2  |-  ( U_ x  e.  A  {
x }  X.  B
)  =  ( A  X.  B )
41, 3eqtr3i 2193 1  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1348   {csn 3583   U_ciun 3873    X. cxp 4609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-iun 3875  df-opab 4051  df-xp 4617
This theorem is referenced by:  ralxp  4754  rexxp  4755  mpompt  5945  mpompts  6177  fmpo  6180  fsumxp  11399  fprodxp  11587  dvfvalap  13444  pwle2  14031  pwf1oexmid  14032
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