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Theorem xpiun 41554
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 41553 . . . . 5 ({𝑥} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}
21eqcomi 2614 . . . 4 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = ({𝑥} × 𝐶)
32a1i 11 . . 3 (𝑥𝐵 → {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = ({𝑥} × 𝐶))
43iuneq2i 4465 . 2 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)} = 𝑥𝐵 ({𝑥} × 𝐶)
5 iunxpconst 5084 . 2 𝑥𝐵 ({𝑥} × 𝐶) = (𝐵 × 𝐶)
64, 5eqtr2i 2628 1 (𝐵 × 𝐶) = 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wcel 1975  {csn 4120   ciun 4445  {copab 4632   × cxp 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-iun 4447  df-opab 4634  df-xp 5030
This theorem is referenced by: (None)
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