Theorem iunxpconst 5741

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 5740	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
2		iunid 5056	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
3	2	xpeq1i 5695	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵)
4	1, 3	eqtr3i 2756	1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)

Syntax hints:   = wceq 1533  {csn 4623  ∪ ciun 4990   × cxp 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-iun 4992  df-opab 5204  df-xp 5675

This theorem is referenced by:  ralxp  5835  rexxp  5836  mpompt  7518  mpompts  8050  fmpo  8053  fsumxp  15724  fprodxp  15932  dvfval  25781  indval2  33542  filnetlem3  35773  sge0xp  45717  xpiun  47108