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Theorem xpiun 46146
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 46145 . . . . 5 ({𝑥} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}
21eqcomi 2742 . . . 4 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = ({𝑥} × 𝐶)
32a1i 11 . . 3 (𝑥𝐵 → {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = ({𝑥} × 𝐶))
43iuneq2i 4976 . 2 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = 𝑥𝐵 ({𝑥} × 𝐶)
5 iunxpconst 5705 . 2 𝑥𝐵 ({𝑥} × 𝐶) = (𝐵 × 𝐶)
64, 5eqtr2i 2762 1 (𝐵 × 𝐶) = 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  {csn 4587   ciun 4955  {copab 5168   × cxp 5632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-iun 4957  df-opab 5169  df-xp 5640
This theorem is referenced by: (None)
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