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Theorem iunxpconst 5761
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 𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iunxpconst
StepHypRef Expression
1 xpiundir 5760 . 2 ( 𝑥𝐴 {𝑥} × 𝐵) = 𝑥𝐴 ({𝑥} × 𝐵)
2 iunid 5065 . . 3 𝑥𝐴 {𝑥} = 𝐴
32xpeq1i 5715 . 2 ( 𝑥𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵)
41, 3eqtr3i 2765 1 𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  {csn 4631   ciun 4996   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-iun 4998  df-opab 5211  df-xp 5695
This theorem is referenced by:  ralxp  5855  rexxp  5856  mpompt  7547  mpompts  8089  fmpo  8092  fsumxp  15805  fprodxp  16015  dvfval  25947  indval2  33995  filnetlem3  36363  sge0xp  46385  xpiun  48002
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