MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunxpconst Structured version   Visualization version   GIF version

Theorem iunxpconst 5757
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 5756 . 2 ( 𝑥𝐴 {𝑥} × 𝐵) = 𝑥𝐴 ({𝑥} × 𝐵)
2 iunid 5059 . . 3 𝑥𝐴 {𝑥} = 𝐴
32xpeq1i 5710 . 2 ( 𝑥𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵)
41, 3eqtr3i 2766 1 𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  {csn 4625   ciun 4990   × cxp 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-iun 4992  df-opab 5205  df-xp 5690
This theorem is referenced by:  ralxp  5851  rexxp  5852  mpompt  7548  mpompts  8091  fmpo  8094  fsumxp  15809  fprodxp  16019  dvfval  25933  indval2  32840  filnetlem3  36382  sge0xp  46449  xpiun  48079
  Copyright terms: Public domain W3C validator