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Theorem br1cossxrnres 37857
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 37812 . . . 4 ((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))
21cosseqi 37836 . . 3 ≀ ((𝑅𝑆) ↾ 𝐴) = ≀ (𝑅 ⋉ (𝑆𝐴))
32breqi 5148 . 2 (⟨𝐵, 𝐶⟩ ≀ ((𝑅𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆𝐴))⟨𝐷, 𝐸⟩)
4 opex 5460 . . . 4 𝐵, 𝐶⟩ ∈ V
5 opex 5460 . . . 4 𝐷, 𝐸⟩ ∈ V
6 br1cossres 37848 . . . 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 37783 . . . . . . 7 ((𝑢 ∈ V ∧ 𝐵𝑉𝐶𝑊) → (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵𝑢𝑆𝐶)))
98el3v1 37627 . . . . . 6 ((𝐵𝑉𝐶𝑊) → (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵𝑢𝑆𝐶)))
10 brxrn 37783 . . . . . . 7 ((𝑢 ∈ V ∧ 𝐷𝑋𝐸𝑌) → (𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷𝑢𝑆𝐸)))
1110el3v1 37627 . . . . . 6 ((𝐷𝑋𝐸𝑌) → (𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷𝑢𝑆𝐸)))
129, 11bi2anan9 637 . . . . 5 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → ((𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩) ↔ ((𝑢𝑅𝐵𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷𝑢𝑆𝐸))))
13 an2anr 635 . . . . 5 (((𝑢𝑅𝐵𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷𝑢𝑆𝐸)) ↔ ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷)))
1412, 13bitrdi 287 . . . 4 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → ((𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩) ↔ ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))
1514rexbidv 3173 . . 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 395  wcel 2099  wrex 3065  Vcvv 3469  cop 4630   class class class wbr 5142  cres 5674  cxrn 37582  ccoss 37583
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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-1st 7987  df-2nd 7988  df-xrn 37780  df-coss 37820
This theorem is referenced by:  br1cossxrnidres  37860  br1cossxrncnvepres  37861  br1cossxrncnvssrres  37917
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