Theorem iunxpconst 5729

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 5728	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
2		iunid 5022	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
3	2	xpeq1i 5682	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵)
4	1, 3	eqtr3i 2788	1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)

This proof depends on syntax axioms:   = wceq 1570  {csn 4588  ∪ ciun 4954   × cxp 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-un 4154  df-sn 4589  df-pr 4591  df-op 4839  df-iun 4956  df-opab 5172  df-xp 5662

This theorem is used by:  ralxp  5825  rexxp  5826  mpompt  7560  mpompts  8104  fmpo  8107  indval2  12355  fsumxp  15970  fprodxp  16181  dvfval  26250  filnetlem3  37359  sge0xp  47654  xpiun  49471