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Theorem br1cossxrnres 38430
Description: 𝐵, 𝐶 and 𝐷, 𝐸 are cosets by an range Cartesian product with a restriction: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.)
Assertion
Ref Expression
br1cossxrnres (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐶   𝑢,𝐷   𝑢,𝐸   𝑢,𝑅   𝑢,𝑆   𝑢,𝑉   𝑢,𝑊   𝑢,𝑋   𝑢,𝑌

Proof of Theorem br1cossxrnres
StepHypRef Expression
1 xrnres2 38385 . . . 4 ((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))
21cosseqi 38409 . . 3 ≀ ((𝑅𝑆) ↾ 𝐴) = ≀ (𝑅 ⋉ (𝑆𝐴))
32breqi 5154 . 2 (⟨𝐵, 𝐶⟩ ≀ ((𝑅𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆𝐴))⟨𝐷, 𝐸⟩)
4 opex 5475 . . . 4 𝐵, 𝐶⟩ ∈ V
5 opex 5475 . . . 4 𝐷, 𝐸⟩ ∈ V
6 br1cossres 38421 . . . 4 ((⟨𝐵, 𝐶⟩ ∈ V ∧ ⟨𝐷, 𝐸⟩ ∈ V) → (⟨𝐵, 𝐶⟩ ≀ ((𝑅𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩)))
74, 5, 6mp2an 692 . . 3 (⟨𝐵, 𝐶⟩ ≀ ((𝑅𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩))
8 brxrn 38356 . . . . . . 7 ((𝑢 ∈ V ∧ 𝐵𝑉𝐶𝑊) → (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵𝑢𝑆𝐶)))
98el3v1 38205 . . . . . 6 ((𝐵𝑉𝐶𝑊) → (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵𝑢𝑆𝐶)))
10 brxrn 38356 . . . . . . 7 ((𝑢 ∈ V ∧ 𝐷𝑋𝐸𝑌) → (𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷𝑢𝑆𝐸)))
1110el3v1 38205 . . . . . 6 ((𝐷𝑋𝐸𝑌) → (𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷𝑢𝑆𝐸)))
129, 11bi2anan9 638 . . . . 5 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → ((𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩) ↔ ((𝑢𝑅𝐵𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷𝑢𝑆𝐸))))
13 an2anr 636 . . . . 5 (((𝑢𝑅𝐵𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷𝑢𝑆𝐸)) ↔ ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷)))
1412, 13bitrdi 287 . . . 4 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → ((𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩) ↔ ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))
1514rexbidv 3177 . . 3 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (∃𝑢𝐴 (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩) ↔ ∃𝑢𝐴 ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))
167, 15bitrid 283 . 2 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ ((𝑅𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))
173, 16bitr3id 285 1 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  wrex 3068  Vcvv 3478  cop 4637   class class class wbr 5148  cres 5691  cxrn 38161  ccoss 38162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-xrn 38353  df-coss 38393
This theorem is referenced by:  br1cossxrnidres  38433  br1cossxrncnvepres  38434  br1cossxrncnvssrres  38490
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