Theorem iunxpconst 5315

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 5314	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
2		iunid 4709	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
3	2	xpeq1i 5275	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵)
4	1, 3	eqtr3i 2795	1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)

Syntax hints:   = wceq 1631  {csn 4316  ∪ ciun 4654   × cxp 5247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-iun 4656  df-opab 4847  df-xp 5255

This theorem is referenced by:  ralxp  5402  rexxp  5403  mpt2mpt  6899  mpt2mpts  7384  fmpt2  7387  fsumxp  14711  fprodxp  14919  dvfval  23881  indval2  30416  filnetlem3  32712  sge0xp  41163  xpiun  42294