Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brcart Structured version   Visualization version   GIF version

Theorem brcart 32376
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 5060 . 2 𝐴, 𝐵⟩ ∈ V
2 brcart.3 . 2 𝐶 ∈ V
3 df-cart 32309 . 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 5306 . . 3 𝐴, 𝐵⟩ ∈ (V × V)
7 brxp 5287 . . 3 (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
86, 2, 7mpbir2an 690 . 2 𝐴, 𝐵⟩((V × V) × V)𝐶
9 3anass 1080 . . . . 5 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 E 𝐴𝑧 E 𝐵)))
104epelc 5164 . . . . . . 7 (𝑦 E 𝐴𝑦𝐴)
115epelc 5164 . . . . . . 7 (𝑧 E 𝐵𝑧𝐵)
1210, 11anbi12i 612 . . . . . 6 ((𝑦 E 𝐴𝑧 E 𝐵) ↔ (𝑦𝐴𝑧𝐵))
1312anbi2i 609 . . . . 5 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 E 𝐴𝑧 E 𝐵)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
149, 13bitri 264 . . . 4 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
15142exbii 1925 . . 3 (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
16 vex 3354 . . . 4 𝑥 ∈ V
1716, 4, 5brpprod3b 32331 . . 3 (𝑥pprod( E , E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴𝑧 E 𝐵))
18 elxp 5271 . . 3 (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
1915, 17, 183bitr4ri 293 . 2 (𝑥 ∈ (𝐴 × 𝐵) ↔ 𝑥pprod( E , E )⟨𝐴, 𝐵⟩)
201, 2, 3, 8, 19brtxpsd3 32340 1 (⟨𝐴, 𝐵⟩Cart𝐶𝐶 = (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  cop 4322   class class class wbr 4786   E cep 5161   × cxp 5247  pprodcpprod 32275  Cartccart 32285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-symdif 3993  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-eprel 5162  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-1st 7315  df-2nd 7316  df-txp 32298  df-pprod 32299  df-cart 32309
This theorem is referenced by:  brimg  32381  brrestrict  32393
  Copyright terms: Public domain W3C validator