Theorem iunxpconst 4720

Description: Membership in a union of cross 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 4719	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
2		iunid 3969	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
3	2	xpeq1i 4680	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵)
4	1, 3	eqtr3i 2216	1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)

Syntax hints:   = wceq 1364  {csn 3619  ∪ ciun 3913   × cxp 4658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-iun 3915  df-opab 4092  df-xp 4666

This theorem is referenced by:  ralxp  4806  rexxp  4807  mpompt  6011  mpompts  6253  fmpo  6256  fsumxp  11582  fprodxp  11770  dvfvalap  14860  pwle2  15559  pwf1oexmid  15560