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Theorem xpiun 47220
Description: A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.)
Assertion
Ref Expression
xpiun (𝐵 × 𝐶) = 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}
Distinct variable groups:   𝑥,𝐵   𝐶,𝑎,𝑏,𝑥
Allowed substitution hints:   𝐵(𝑎,𝑏)

Proof of Theorem xpiun
StepHypRef Expression
1 xpsnopab 47219 . . . . 5 ({𝑥} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}
21eqcomi 2737 . . . 4 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = ({𝑥} × 𝐶)
32a1i 11 . . 3 (𝑥𝐵 → {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = ({𝑥} × 𝐶))
43iuneq2i 5017 . 2 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = 𝑥𝐵 ({𝑥} × 𝐶)
5 iunxpconst 5750 . 2 𝑥𝐵 ({𝑥} × 𝐶) = (𝐵 × 𝐶)
64, 5eqtr2i 2757 1 (𝐵 × 𝐶) = 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wcel 2099  {csn 4629   ciun 4996  {copab 5210   × cxp 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iun 4998  df-opab 5211  df-xp 5684
This theorem is referenced by: (None)
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