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Theorem xpiun 42294
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 42293 . . . . 5 ({𝑥} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}
21eqcomi 2769 . . . 4 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = ({𝑥} × 𝐶)
32a1i 11 . . 3 (𝑥𝐵 → {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = ({𝑥} × 𝐶))
43iuneq2i 4691 . 2 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = 𝑥𝐵 ({𝑥} × 𝐶)
5 iunxpconst 5332 . 2 𝑥𝐵 ({𝑥} × 𝐶) = (𝐵 × 𝐶)
64, 5eqtr2i 2783 1 (𝐵 × 𝐶) = 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wcel 2139  {csn 4321   ciun 4672  {copab 4864   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-iun 4674  df-opab 4865  df-xp 5272
This theorem is referenced by: (None)
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