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Theorem br1cossxrncnvssrres 35901
Description: 𝐵, 𝐶 and 𝐷, 𝐸 are cosets by range 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 35841 . 2 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( S ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢 S 𝐶𝑢𝑅𝐵) ∧ (𝑢 S 𝐸𝑢𝑅𝐷))))
2 brcnvssr 35899 . . . . . 6 (𝑢 ∈ V → (𝑢 S 𝐶𝐶𝑢))
32elv 3449 . . . . 5 (𝑢 S 𝐶𝐶𝑢)
43anbi1i 626 . . . 4 ((𝑢 S 𝐶𝑢𝑅𝐵) ↔ (𝐶𝑢𝑢𝑅𝐵))
5 brcnvssr 35899 . . . . . 6 (𝑢 ∈ V → (𝑢 S 𝐸𝐸𝑢))
65elv 3449 . . . . 5 (𝑢 S 𝐸𝐸𝑢)
76anbi1i 626 . . . 4 ((𝑢 S 𝐸𝑢𝑅𝐷) ↔ (𝐸𝑢𝑢𝑅𝐷))
84, 7anbi12i 629 . . 3 (((𝑢 S 𝐶𝑢𝑅𝐵) ∧ (𝑢 S 𝐸𝑢𝑅𝐷)) ↔ ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷)))
98rexbii 3213 . 2 (∃𝑢𝐴 ((𝑢 S 𝐶𝑢𝑅𝐵) ∧ (𝑢 S 𝐸𝑢𝑅𝐷)) ↔ ∃𝑢𝐴 ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷)))
101, 9syl6bb 290 1 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( S ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2112  wrex 3110  Vcvv 3444  wss 3884  cop 4534   class class class wbr 5033  ccnv 5522  cres 5525  cxrn 35605  ccoss 35606   S cssr 35609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-1st 7675  df-2nd 7676  df-xrn 35776  df-coss 35812  df-ssr 35891
This theorem is referenced by: (None)
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