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Theorem inxpxrn 36288
Description: Two ways to express the intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 10-Apr-2020.)
Assertion
Ref Expression
inxpxrn ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶))) = ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))

Proof of Theorem inxpxrn
Dummy variables 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrnrel 36270 . 2 Rel ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))
2 relinxp 5699 . 2 Rel ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))
3 brxrn2 36272 . . . . . 6 (𝑢 ∈ V → (𝑢(𝑅𝑆)𝑥 ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)))
43elv 3427 . . . . 5 (𝑢(𝑅𝑆)𝑥 ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))
54anbi2i 626 . . . 4 ((𝑢𝐴𝑢(𝑅𝑆)𝑥) ↔ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)))
65anbi2i 626 . . 3 ((𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
7 xrninxp2 36286 . . . 4 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
87brabidgaw 36262 . . 3 (𝑢((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))𝑥 ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)))
9 brxrn2 36272 . . . . 5 (𝑢 ∈ V → (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥 ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)))
109elv 3427 . . . 4 (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥 ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧))
11 3anass 1097 . . . . 5 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)))
12112exbii 1856 . . . 4 (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)))
13 brinxp2 5641 . . . . . . . . . . . 12 (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑢𝐴𝑦𝐵) ∧ 𝑢𝑅𝑦))
14 brinxp2 5641 . . . . . . . . . . . 12 (𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧 ↔ ((𝑢𝐴𝑧𝐶) ∧ 𝑢𝑆𝑧))
1513, 14anbi12i 630 . . . . . . . . . . 11 ((𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ (((𝑢𝐴𝑦𝐵) ∧ 𝑢𝑅𝑦) ∧ ((𝑢𝐴𝑧𝐶) ∧ 𝑢𝑆𝑧)))
16 anan 36146 . . . . . . . . . . 11 ((((𝑢𝐴𝑦𝐵) ∧ 𝑢𝑅𝑦) ∧ ((𝑢𝐴𝑧𝐶) ∧ 𝑢𝑆𝑧)) ↔ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
1715, 16bitri 278 . . . . . . . . . 10 ((𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
1817anbi2i 626 . . . . . . . . 9 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))))
19 anass 472 . . . . . . . . 9 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))))
20 eqelb 36149 . . . . . . . . . . 11 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶)))
21 opelxp 5602 . . . . . . . . . . . 12 (⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶) ↔ (𝑦𝐵𝑧𝐶))
2221anbi2i 626 . . . . . . . . . . 11 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧𝐶)))
2320, 22bitr2i 279 . . . . . . . . . 10 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)))
2423anbi1i 627 . . . . . . . . 9 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
2518, 19, 243bitr2i 302 . . . . . . . 8 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
26 ancom 464 . . . . . . . . 9 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩))
2726anbi1i 627 . . . . . . . 8 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ ((𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
28 anass 472 . . . . . . . 8 (((𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))))
2925, 27, 283bitri 300 . . . . . . 7 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))))
30 an12 645 . . . . . . . . 9 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
31 3anass 1097 . . . . . . . . . 10 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))
3231anbi2i 626 . . . . . . . . 9 ((𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)) ↔ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
3330, 32bitr4i 281 . . . . . . . 8 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)))
3433anbi2i 626 . . . . . . 7 ((𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
3529, 34bitri 278 . . . . . 6 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
36352exbii 1856 . . . . 5 (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ ∃𝑦𝑧(𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
37 19.42vv 1966 . . . . 5 (∃𝑦𝑧(𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ ∃𝑦𝑧(𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
38 19.42vv 1966 . . . . . 6 (∃𝑦𝑧(𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)) ↔ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)))
3938anbi2i 626 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐶) ∧ ∃𝑦𝑧(𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
4036, 37, 393bitri 300 . . . 4 (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
4110, 12, 403bitri 300 . . 3 (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥 ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
426, 8, 413bitr4ri 307 . 2 (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥𝑢((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))𝑥)
431, 2, 42eqbrriv 5676 1 ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶))) = ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2111  Vcvv 3421  cin 3880  cop 4562   class class class wbr 5068   × cxp 5564  cxrn 36099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337  ax-un 7542
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-opab 5131  df-mpt 5151  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-fo 6404  df-fv 6406  df-1st 7780  df-2nd 7781  df-xrn 36268
This theorem is referenced by:  xrnres4  36298  xrnresex  36299
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