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Theorem dfxp3 7994
Description: Define the Cartesian product of three classes. Compare df-xp 5640. (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 7988 . 2 {⟨𝑢, 𝑧⟩ ∣ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑧𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶)}
3 df-xp 5640 . 2 ((𝐴 × 𝐵) × 𝐶) = {⟨𝑢, 𝑧⟩ ∣ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑧𝐶)}
4 df-3an 1090 . . 3 ((𝑥𝐴𝑦𝐵𝑧𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶))
54oprabbii 7425 . 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 397  w3a 1088   = wceq 1542  wcel 2107  cop 4593  {copab 5168   × cxp 5632  {coprab 7359
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  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-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-oprab 7362  df-1st 7922  df-2nd 7923
This theorem is referenced by: (None)
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