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Theorem iunxpconst 5744
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 5743 . 2 ( 𝑥𝐴 {𝑥} × 𝐵) = 𝑥𝐴 ({𝑥} × 𝐵)
2 iunid 5058 . . 3 𝑥𝐴 {𝑥} = 𝐴
32xpeq1i 5698 . 2 ( 𝑥𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵)
41, 3eqtr3i 2755 1 𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  {csn 4624   ciun 4991   × cxp 5670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-iun 4993  df-opab 5206  df-xp 5678
This theorem is referenced by:  ralxp  5838  rexxp  5839  mpompt  7531  mpompts  8067  fmpo  8070  fsumxp  15750  fprodxp  15958  dvfval  25844  indval2  33690  filnetlem3  35921  sge0xp  45880  xpiun  47332
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