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Theorem iunxpconst 5330
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 5329 . 2 ( 𝑥𝐴 {𝑥} × 𝐵) = 𝑥𝐴 ({𝑥} × 𝐵)
2 iunid 4725 . . 3 𝑥𝐴 {𝑥} = 𝐴
32xpeq1i 5290 . 2 ( 𝑥𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵)
41, 3eqtr3i 2782 1 𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1630  {csn 4319   ciun 4670   × cxp 5262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-iun 4672  df-opab 4863  df-xp 5270
This theorem is referenced by:  ralxp  5417  rexxp  5418  mpt2mpt  6915  mpt2mpts  7400  fmpt2  7403  fsumxp  14700  fprodxp  14909  dvfval  23858  indval2  30383  filnetlem3  32679  sge0xp  41147  xpiun  42274
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