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Theorem br1cossxrnres 37256
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 37211 . . . 4 ((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))
21cosseqi 37235 . . 3 ≀ ((𝑅𝑆) ↾ 𝐴) = ≀ (𝑅 ⋉ (𝑆𝐴))
32breqi 5153 . 2 (⟨𝐵, 𝐶⟩ ≀ ((𝑅𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆𝐴))⟨𝐷, 𝐸⟩)
4 opex 5463 . . . 4 𝐵, 𝐶⟩ ∈ V
5 opex 5463 . . . 4 𝐷, 𝐸⟩ ∈ V
6 br1cossres 37247 . . . 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 691 . . 3 (⟨𝐵, 𝐶⟩ ≀ ((𝑅𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩))
8 brxrn 37182 . . . . . . 7 ((𝑢 ∈ V ∧ 𝐵𝑉𝐶𝑊) → (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵𝑢𝑆𝐶)))
98el3v1 37025 . . . . . 6 ((𝐵𝑉𝐶𝑊) → (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵𝑢𝑆𝐶)))
10 brxrn 37182 . . . . . . 7 ((𝑢 ∈ V ∧ 𝐷𝑋𝐸𝑌) → (𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷𝑢𝑆𝐸)))
1110el3v1 37025 . . . . . 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 3179 . . 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 205  wa 397  wcel 2107  wrex 3071  Vcvv 3475  cop 4633   class class class wbr 5147  cres 5677  cxrn 36980  ccoss 36981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-1st 7970  df-2nd 7971  df-xrn 37179  df-coss 37219
This theorem is referenced by:  br1cossxrnidres  37259  br1cossxrncnvepres  37260  br1cossxrncnvssrres  37316
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