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Theorem iunxpconst 5141
 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 5140 . 2 ( 𝑥𝐴 {𝑥} × 𝐵) = 𝑥𝐴 ({𝑥} × 𝐵)
2 iunid 4546 . . 3 𝑥𝐴 {𝑥} = 𝐴
32xpeq1i 5100 . 2 ( 𝑥𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵)
41, 3eqtr3i 2645 1 𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480  {csn 4153  ∪ ciun 4490   × cxp 5077 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-iun 4492  df-opab 4679  df-xp 5085 This theorem is referenced by:  ralxp  5228  rexxp  5229  mpt2mpt  6712  mpt2mpts  7186  fmpt2  7189  fsumxp  14442  fprodxp  14648  dvfval  23584  indval2  29882  filnetlem3  32052  sge0xp  39979  xpiun  41080
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