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Theorem inxpxrn 34476
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 34458 . 2 Rel ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))
2 relinxp 34393 . 2 Rel ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))
3 brxrn2 34460 . . . . . 6 (𝑢 ∈ V → (𝑢(𝑅𝑆)𝑥 ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)))
43elv 34309 . . . . 5 (𝑢(𝑅𝑆)𝑥 ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))
54anbi2i 732 . . . 4 ((𝑢𝐴𝑢(𝑅𝑆)𝑥) ↔ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)))
65anbi2i 732 . . 3 ((𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
7 xrninxp2 34474 . . . 4 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
87brabidga 34451 . . 3 (𝑢((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))𝑥 ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)))
9 brxrn2 34460 . . . . 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 34309 . . . 4 (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥 ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧))
11 3anass 1081 . . . . 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 1924 . . . 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 brinxp2ALTV 34358 . . . . . . . . . . . 12 (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑢𝐴𝑦𝐵) ∧ 𝑢𝑅𝑦))
14 brinxp2ALTV 34358 . . . . . . . . . . . 12 (𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧 ↔ ((𝑢𝐴𝑧𝐶) ∧ 𝑢𝑆𝑧))
1513, 14anbi12i 735 . . . . . . . . . . 11 ((𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ (((𝑢𝐴𝑦𝐵) ∧ 𝑢𝑅𝑦) ∧ ((𝑢𝐴𝑧𝐶) ∧ 𝑢𝑆𝑧)))
16 anan 34325 . . . . . . . . . . 11 ((((𝑢𝐴𝑦𝐵) ∧ 𝑢𝑅𝑦) ∧ ((𝑢𝐴𝑧𝐶) ∧ 𝑢𝑆𝑧)) ↔ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
1715, 16bitri 264 . . . . . . . . . 10 ((𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
1817anbi2i 732 . . . . . . . . 9 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))))
19 anass 684 . . . . . . . . 9 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))))
20 eqelb 34328 . . . . . . . . . . 11 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶)))
21 opelxp 5303 . . . . . . . . . . . 12 (⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶) ↔ (𝑦𝐵𝑧𝐶))
2221anbi2i 732 . . . . . . . . . . 11 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧𝐶)))
2320, 22bitr2i 265 . . . . . . . . . 10 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)))
2423anbi1i 733 . . . . . . . . 9 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
2518, 19, 243bitr2i 288 . . . . . . . 8 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
26 ancom 465 . . . . . . . . 9 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩))
2726anbi1i 733 . . . . . . . 8 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ ((𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
28 anass 684 . . . . . . . 8 (((𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))))
2925, 27, 283bitri 286 . . . . . . 7 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))))
30 an12 873 . . . . . . . . 9 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
31 3anass 1081 . . . . . . . . . 10 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))
3231anbi2i 732 . . . . . . . . 9 ((𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)) ↔ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
3330, 32bitr4i 267 . . . . . . . 8 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)))
3433anbi2i 732 . . . . . . 7 ((𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
3529, 34bitri 264 . . . . . 6 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
36352exbii 1924 . . . . 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 2032 . . . . 5 (∃𝑦𝑧(𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ ∃𝑦𝑧(𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
38 19.42vv 2032 . . . . . 6 (∃𝑦𝑧(𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)) ↔ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)))
3938anbi2i 732 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐶) ∧ ∃𝑦𝑧(𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
4036, 37, 393bitri 286 . . . 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 286 . . 3 (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥 ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
426, 8, 413bitr4ri 293 . 2 (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥𝑢((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))𝑥)
431, 2, 42eqbrriv 5372 1 ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶))) = ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  cin 3714  cop 4327   class class class wbr 4804   × cxp 5264  cxrn 34295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-1st 7333  df-2nd 7334  df-xrn 34456
This theorem is referenced by:  xrnres4  34486  xrnresex  34487
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