Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inxpxrn Structured version   Visualization version   GIF version

Theorem inxpxrn 35635
 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 35617 . 2 Rel ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))
2 relinxp 5680 . 2 Rel ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))
3 brxrn2 35619 . . . . . 6 (𝑢 ∈ V → (𝑢(𝑅𝑆)𝑥 ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)))
43elv 3498 . . . . 5 (𝑢(𝑅𝑆)𝑥 ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))
54anbi2i 624 . . . 4 ((𝑢𝐴𝑢(𝑅𝑆)𝑥) ↔ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)))
65anbi2i 624 . . 3 ((𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
7 xrninxp2 35633 . . . 4 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
87brabidgaw 35609 . . 3 (𝑢((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))𝑥 ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)))
9 brxrn2 35619 . . . . 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 3498 . . . 4 (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥 ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧))
11 3anass 1090 . . . . 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 1843 . . . 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 5622 . . . . . . . . . . . 12 (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑢𝐴𝑦𝐵) ∧ 𝑢𝑅𝑦))
14 brinxp2 5622 . . . . . . . . . . . 12 (𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧 ↔ ((𝑢𝐴𝑧𝐶) ∧ 𝑢𝑆𝑧))
1513, 14anbi12i 628 . . . . . . . . . . 11 ((𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ (((𝑢𝐴𝑦𝐵) ∧ 𝑢𝑅𝑦) ∧ ((𝑢𝐴𝑧𝐶) ∧ 𝑢𝑆𝑧)))
16 anan 35491 . . . . . . . . . . 11 ((((𝑢𝐴𝑦𝐵) ∧ 𝑢𝑅𝑦) ∧ ((𝑢𝐴𝑧𝐶) ∧ 𝑢𝑆𝑧)) ↔ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
1715, 16bitri 277 . . . . . . . . . 10 ((𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
1817anbi2i 624 . . . . . . . . 9 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))))
19 anass 471 . . . . . . . . 9 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))))
20 eqelb 35494 . . . . . . . . . . 11 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶)))
21 opelxp 5584 . . . . . . . . . . . 12 (⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶) ↔ (𝑦𝐵𝑧𝐶))
2221anbi2i 624 . . . . . . . . . . 11 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧𝐶)))
2320, 22bitr2i 278 . . . . . . . . . 10 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)))
2423anbi1i 625 . . . . . . . . 9 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
2518, 19, 243bitr2i 301 . . . . . . . 8 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
26 ancom 463 . . . . . . . . 9 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩))
2726anbi1i 625 . . . . . . . 8 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ ((𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
28 anass 471 . . . . . . . 8 (((𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))))
2925, 27, 283bitri 299 . . . . . . 7 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))))
30 an12 643 . . . . . . . . 9 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
31 3anass 1090 . . . . . . . . . 10 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))
3231anbi2i 624 . . . . . . . . 9 ((𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)) ↔ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))))
3330, 32bitr4i 280 . . . . . . . 8 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)))
3433anbi2i 624 . . . . . . 7 ((𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝐴 ∧ (𝑢𝑅𝑦𝑢𝑆𝑧)))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
3529, 34bitri 277 . . . . . 6 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
36352exbii 1843 . . . . 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 1952 . . . . 5 (∃𝑦𝑧(𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ ∃𝑦𝑧(𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
38 19.42vv 1952 . . . . . 6 (∃𝑦𝑧(𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)) ↔ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧)))
3938anbi2i 624 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐶) ∧ ∃𝑦𝑧(𝑢𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
4036, 37, 393bitri 299 . . . 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 299 . . 3 (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥 ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴 ∧ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦𝑢𝑆𝑧))))
426, 8, 413bitr4ri 306 . 2 (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥𝑢((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))𝑥)
431, 2, 42eqbrriv 5657 1 ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶))) = ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1531  ∃wex 1774   ∈ wcel 2108  Vcvv 3493   ∩ cin 3933  ⟨cop 4565   class class class wbr 5057   × cxp 5546   ⋉ cxrn 35444 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7681  df-2nd 7682  df-xrn 35615 This theorem is referenced by:  xrnres4  35645  xrnresex  35646
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