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Theorem br1cossxrncnvssrres 34581
 Description: ⟨𝐵, 𝐶⟩ and ⟨𝐷, 𝐸⟩ are cosets by tail Cartesian product with restricted converse subsets class: a binary relation. (Contributed by Peter Mazsa, 9-Jun-2021.)
Assertion
Ref Expression
br1cossxrncnvssrres (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( S ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷))))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐶   𝑢,𝐷   𝑢,𝐸   𝑢,𝑅   𝑢,𝑉   𝑢,𝑊   𝑢,𝑋   𝑢,𝑌

Proof of Theorem br1cossxrncnvssrres
StepHypRef Expression
1 br1cossxrnres 34521 . 2 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( S ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢 S 𝐶𝑢𝑅𝐵) ∧ (𝑢 S 𝐸𝑢𝑅𝐷))))
2 brcnvssr 34579 . . . . . 6 (𝑢 ∈ V → (𝑢 S 𝐶𝐶𝑢))
32elv 34309 . . . . 5 (𝑢 S 𝐶𝐶𝑢)
43anbi1i 733 . . . 4 ((𝑢 S 𝐶𝑢𝑅𝐵) ↔ (𝐶𝑢𝑢𝑅𝐵))
5 brcnvssr 34579 . . . . . 6 (𝑢 ∈ V → (𝑢 S 𝐸𝐸𝑢))
65elv 34309 . . . . 5 (𝑢 S 𝐸𝐸𝑢)
76anbi1i 733 . . . 4 ((𝑢 S 𝐸𝑢𝑅𝐷) ↔ (𝐸𝑢𝑢𝑅𝐷))
84, 7anbi12i 735 . . 3 (((𝑢 S 𝐶𝑢𝑅𝐵) ∧ (𝑢 S 𝐸𝑢𝑅𝐷)) ↔ ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷)))
98rexbii 3179 . 2 (∃𝑢𝐴 ((𝑢 S 𝐶𝑢𝑅𝐵) ∧ (𝑢 S 𝐸𝑢𝑅𝐷)) ↔ ∃𝑢𝐴 ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷)))
101, 9syl6bb 276 1 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( S ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∈ wcel 2139  ∃wrex 3051  Vcvv 3340   ⊆ wss 3715  ⟨cop 4327   class class class wbr 4804  ◡ccnv 5265   ↾ cres 5268   ⋉ cxrn 34295   ≀ ccoss 34296   S cssr 34299 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  df-coss 34492  df-ssr 34571 This theorem is referenced by: (None)
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