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Theorem xpiun 48002
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 48001 . . . . 5 ({𝑥} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}
21eqcomi 2744 . . . 4 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = ({𝑥} × 𝐶)
32a1i 11 . . 3 (𝑥𝐵 → {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = ({𝑥} × 𝐶))
43iuneq2i 5018 . 2 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = 𝑥𝐵 ({𝑥} × 𝐶)
5 iunxpconst 5761 . 2 𝑥𝐵 ({𝑥} × 𝐶) = (𝐵 × 𝐶)
64, 5eqtr2i 2764 1 (𝐵 × 𝐶) = 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  {csn 4631   ciun 4996  {copab 5210   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-iun 4998  df-opab 5211  df-xp 5695
This theorem is referenced by: (None)
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