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Theorem brcart 34484
Description: Binary relation form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brcart.1 𝐴 ∈ V
brcart.2 𝐵 ∈ V
brcart.3 𝐶 ∈ V
Assertion
Ref Expression
brcart (⟨𝐴, 𝐵⟩Cart𝐶𝐶 = (𝐴 × 𝐵))

Proof of Theorem brcart
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 5419 . 2 𝐴, 𝐵⟩ ∈ V
2 brcart.3 . 2 𝐶 ∈ V
3 df-cart 34417 . 2 Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
4 brcart.1 . . . 4 𝐴 ∈ V
5 brcart.2 . . . 4 𝐵 ∈ V
64, 5opelvv 5670 . . 3 𝐴, 𝐵⟩ ∈ (V × V)
7 brxp 5679 . . 3 (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
86, 2, 7mpbir2an 709 . 2 𝐴, 𝐵⟩((V × V) × V)𝐶
9 3anass 1095 . . . . 5 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 E 𝐴𝑧 E 𝐵)))
104epeli 5537 . . . . . . 7 (𝑦 E 𝐴𝑦𝐴)
115epeli 5537 . . . . . . 7 (𝑧 E 𝐵𝑧𝐵)
1210, 11anbi12i 627 . . . . . 6 ((𝑦 E 𝐴𝑧 E 𝐵) ↔ (𝑦𝐴𝑧𝐵))
1312anbi2i 623 . . . . 5 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 E 𝐴𝑧 E 𝐵)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
149, 13bitri 274 . . . 4 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
15142exbii 1851 . . 3 (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
16 vex 3447 . . . 4 𝑥 ∈ V
1716, 4, 5brpprod3b 34439 . . 3 (𝑥pprod( E , E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵))
18 elxp 5654 . . 3 (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
1915, 17, 183bitr4ri 303 . 2 (𝑥 ∈ (𝐴 × 𝐵) ↔ 𝑥pprod( E , E )⟨𝐴, 𝐵⟩)
201, 2, 3, 8, 19brtxpsd3 34448 1 (⟨𝐴, 𝐵⟩Cart𝐶𝐶 = (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  Vcvv 3443  cop 4590   class class class wbr 5103   E cep 5534   × cxp 5629  pprodcpprod 34383  Cartccart 34393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-symdif 4200  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-eprel 5535  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7917  df-2nd 7918  df-txp 34406  df-pprod 34407  df-cart 34417
This theorem is referenced by:  brimg  34489  brrestrict  34501
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