Theorem iunxpconst 5732

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	⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iunxpconst

Step	Hyp	Ref	Expression
1		xpiundir 5731	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
2		iunid 5041	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
3	2	xpeq1i 5685	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵)
4	1, 3	eqtr3i 2761	1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)

Syntax hints:   = wceq 1540  {csn 4606  ∪ ciun 4972   × cxp 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-iun 4974  df-opab 5187  df-xp 5665

This theorem is referenced by:  ralxp  5826  rexxp  5827  mpompt  7526  mpompts  8069  fmpo  8072  fsumxp  15793  fprodxp  16003  dvfval  25855  indval2  32836  filnetlem3  36403  sge0xp  46425  xpiun  48100