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Theorem brcart 35913
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 5474 . 2 𝐴, 𝐵⟩ ∈ V
2 brcart.3 . 2 𝐶 ∈ V
3 df-cart 35846 . 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 5728 . . 3 𝐴, 𝐵⟩ ∈ (V × V)
7 brxp 5737 . . 3 (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
86, 2, 7mpbir2an 711 . 2 𝐴, 𝐵⟩((V × V) × V)𝐶
9 3anass 1094 . . . . 5 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 E 𝐴𝑧 E 𝐵)))
104epeli 5590 . . . . . . 7 (𝑦 E 𝐴𝑦𝐴)
115epeli 5590 . . . . . . 7 (𝑧 E 𝐵𝑧𝐵)
1210, 11anbi12i 628 . . . . . 6 ((𝑦 E 𝐴𝑧 E 𝐵) ↔ (𝑦𝐴𝑧𝐵))
1312anbi2i 623 . . . . 5 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 E 𝐴𝑧 E 𝐵)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
149, 13bitri 275 . . . 4 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
15142exbii 1845 . . 3 (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
16 vex 3481 . . . 4 𝑥 ∈ V
1716, 4, 5brpprod3b 35868 . . 3 (𝑥pprod( E , E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵))
18 elxp 5711 . . 3 (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
1915, 17, 183bitr4ri 304 . 2 (𝑥 ∈ (𝐴 × 𝐵) ↔ 𝑥pprod( E , E )⟨𝐴, 𝐵⟩)
201, 2, 3, 8, 19brtxpsd3 35877 1 (⟨𝐴, 𝐵⟩Cart𝐶𝐶 = (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  Vcvv 3477  cop 4636   class class class wbr 5147   E cep 5587   × cxp 5686  pprodcpprod 35812  Cartccart 35822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-symdif 4258  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-eprel 5588  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-1st 8012  df-2nd 8013  df-txp 35835  df-pprod 35836  df-cart 35846
This theorem is referenced by:  brimg  35918  brrestrict  35930
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