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Theorem dfxp3 8086
Description: Define the Cartesian product of three classes. Compare df-xp 5691. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
dfxp3 ((𝐴 × 𝐵) × 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥𝐴𝑦𝐵𝑧𝐶)}
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧

Proof of Theorem dfxp3
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 biidd 262 . . 3 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑧𝐶𝑧𝐶))
21dfoprab4 8080 . 2 {⟨𝑢, 𝑧⟩ ∣ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑧𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶)}
3 df-xp 5691 . 2 ((𝐴 × 𝐵) × 𝐶) = {⟨𝑢, 𝑧⟩ ∣ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑧𝐶)}
4 df-3an 1089 . . 3 ((𝑥𝐴𝑦𝐵𝑧𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶))
54oprabbii 7500 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥𝐴𝑦𝐵𝑧𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶)}
62, 3, 53eqtr4i 2775 1 ((𝐴 × 𝐵) × 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥𝐴𝑦𝐵𝑧𝐶)}
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1087   = wceq 1540  wcel 2108  cop 4632  {copab 5205   × cxp 5683  {coprab 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-oprab 7435  df-1st 8014  df-2nd 8015
This theorem is referenced by:  mpoaddf  11249  mpomulf  11250
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