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Theorem brcart 35933
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 5469 . 2 𝐴, 𝐵⟩ ∈ V
2 brcart.3 . 2 𝐶 ∈ V
3 df-cart 35866 . 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 5725 . . 3 𝐴, 𝐵⟩ ∈ (V × V)
7 brxp 5734 . . 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 1095 . . . . 5 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 E 𝐴𝑧 E 𝐵)))
104epeli 5586 . . . . . . 7 (𝑦 E 𝐴𝑦𝐴)
115epeli 5586 . . . . . . 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 1849 . . 3 (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
16 vex 3484 . . . 4 𝑥 ∈ V
1716, 4, 5brpprod3b 35888 . . 3 (𝑥pprod( E , E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵))
18 elxp 5708 . . 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 35897 1 (⟨𝐴, 𝐵⟩Cart𝐶𝐶 = (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cop 4632   class class class wbr 5143   E cep 5583   × cxp 5683  pprodcpprod 35832  Cartccart 35842
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-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-symdif 4253  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-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-txp 35855  df-pprod 35856  df-cart 35866
This theorem is referenced by:  brimg  35938  brrestrict  35950
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