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Theorem brcart 34563
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 𝐡 ∈ V
brcart.3 𝐢 ∈ V
Assertion
Ref Expression
brcart (⟨𝐴, 𝐡⟩Cart𝐢 ↔ 𝐢 = (𝐴 Γ— 𝐡))

Proof of Theorem brcart
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 5422 . 2 ⟨𝐴, 𝐡⟩ ∈ V
2 brcart.3 . 2 𝐢 ∈ V
3 df-cart 34496 . 2 Cart = (((V Γ— V) Γ— V) βˆ– ran ((V βŠ— E ) β–³ (pprod( E , E ) βŠ— V)))
4 brcart.1 . . . 4 𝐴 ∈ V
5 brcart.2 . . . 4 𝐡 ∈ V
64, 5opelvv 5673 . . 3 ⟨𝐴, 𝐡⟩ ∈ (V Γ— V)
7 brxp 5682 . . 3 (⟨𝐴, 𝐡⟩((V Γ— V) Γ— V)𝐢 ↔ (⟨𝐴, 𝐡⟩ ∈ (V Γ— V) ∧ 𝐢 ∈ V))
86, 2, 7mpbir2an 710 . 2 ⟨𝐴, 𝐡⟩((V Γ— V) Γ— V)𝐢
9 3anass 1096 . . . . 5 ((π‘₯ = βŸ¨π‘¦, π‘§βŸ© ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐡) ↔ (π‘₯ = βŸ¨π‘¦, π‘§βŸ© ∧ (𝑦 E 𝐴 ∧ 𝑧 E 𝐡)))
104epeli 5540 . . . . . . 7 (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)
115epeli 5540 . . . . . . 7 (𝑧 E 𝐡 ↔ 𝑧 ∈ 𝐡)
1210, 11anbi12i 628 . . . . . 6 ((𝑦 E 𝐴 ∧ 𝑧 E 𝐡) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐡))
1312anbi2i 624 . . . . 5 ((π‘₯ = βŸ¨π‘¦, π‘§βŸ© ∧ (𝑦 E 𝐴 ∧ 𝑧 E 𝐡)) ↔ (π‘₯ = βŸ¨π‘¦, π‘§βŸ© ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐡)))
149, 13bitri 275 . . . 4 ((π‘₯ = βŸ¨π‘¦, π‘§βŸ© ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐡) ↔ (π‘₯ = βŸ¨π‘¦, π‘§βŸ© ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐡)))
15142exbii 1852 . . 3 (βˆƒπ‘¦βˆƒπ‘§(π‘₯ = βŸ¨π‘¦, π‘§βŸ© ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐡) ↔ βˆƒπ‘¦βˆƒπ‘§(π‘₯ = βŸ¨π‘¦, π‘§βŸ© ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐡)))
16 vex 3448 . . . 4 π‘₯ ∈ V
1716, 4, 5brpprod3b 34518 . . 3 (π‘₯pprod( E , E )⟨𝐴, 𝐡⟩ ↔ βˆƒπ‘¦βˆƒπ‘§(π‘₯ = βŸ¨π‘¦, π‘§βŸ© ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐡))
18 elxp 5657 . . 3 (π‘₯ ∈ (𝐴 Γ— 𝐡) ↔ βˆƒπ‘¦βˆƒπ‘§(π‘₯ = βŸ¨π‘¦, π‘§βŸ© ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐡)))
1915, 17, 183bitr4ri 304 . 2 (π‘₯ ∈ (𝐴 Γ— 𝐡) ↔ π‘₯pprod( E , E )⟨𝐴, 𝐡⟩)
201, 2, 3, 8, 19brtxpsd3 34527 1 (⟨𝐴, 𝐡⟩Cart𝐢 ↔ 𝐢 = (𝐴 Γ— 𝐡))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3444  βŸ¨cop 4593   class class class wbr 5106   E cep 5537   Γ— cxp 5632  pprodcpprod 34462  Cartccart 34472
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 2704  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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-symdif 4203  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-eprel 5538  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-1st 7922  df-2nd 7923  df-txp 34485  df-pprod 34486  df-cart 34496
This theorem is referenced by:  brimg  34568  brrestrict  34580
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