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Theorem br1cossxrncnvepres 37789
Description: 𝐵, 𝐶 and 𝐷, 𝐸 are cosets by a range Cartesian product with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 12-May-2021.)
Assertion
Ref Expression
br1cossxrncnvepres (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( E ↾ 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 br1cossxrncnvepres
StepHypRef Expression
1 br1cossxrnres 37785 . 2 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( E ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢 E 𝐶𝑢𝑅𝐵) ∧ (𝑢 E 𝐸𝑢𝑅𝐷))))
2 brcnvep 37600 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝐶𝐶𝑢))
32elv 3479 . . . . 5 (𝑢 E 𝐶𝐶𝑢)
43anbi1i 623 . . . 4 ((𝑢 E 𝐶𝑢𝑅𝐵) ↔ (𝐶𝑢𝑢𝑅𝐵))
5 brcnvep 37600 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝐸𝐸𝑢))
65elv 3479 . . . . 5 (𝑢 E 𝐸𝐸𝑢)
76anbi1i 623 . . . 4 ((𝑢 E 𝐸𝑢𝑅𝐷) ↔ (𝐸𝑢𝑢𝑅𝐷))
84, 7anbi12i 626 . . 3 (((𝑢 E 𝐶𝑢𝑅𝐵) ∧ (𝑢 E 𝐸𝑢𝑅𝐷)) ↔ ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷)))
98rexbii 3093 . 2 (∃𝑢𝐴 ((𝑢 E 𝐶𝑢𝑅𝐵) ∧ (𝑢 E 𝐸𝑢𝑅𝐷)) ↔ ∃𝑢𝐴 ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷)))
101, 9bitrdi 287 1 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( E ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2105  wrex 3069  Vcvv 3473  cop 4634   class class class wbr 5148   E cep 5579  ccnv 5675  cres 5678  cxrn 37509  ccoss 37510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7979  df-2nd 7980  df-xrn 37708  df-coss 37748
This theorem is referenced by: (None)
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