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Theorem iunxpconst 5421
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 5420 . 2 ( 𝑥𝐴 {𝑥} × 𝐵) = 𝑥𝐴 ({𝑥} × 𝐵)
2 iunid 4808 . . 3 𝑥𝐴 {𝑥} = 𝐴
32xpeq1i 5381 . 2 ( 𝑥𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵)
41, 3eqtr3i 2804 1 𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  {csn 4398   ciun 4753   × cxp 5353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-iun 4755  df-opab 4949  df-xp 5361
This theorem is referenced by:  ralxp  5509  rexxp  5510  mpt2mpt  7029  mpt2mpts  7514  fmpt2  7517  fsumxp  14908  fprodxp  15115  dvfval  24098  indval2  30674  filnetlem3  32963  sge0xp  41570  xpiun  42781
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