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Theorem br1cossxrnidres 38432
Description: 𝐵, 𝐶 and 𝐷, 𝐸 are cosets by a range Cartesian product with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.)
Assertion
Ref Expression
br1cossxrnidres (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( I ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢 = 𝐶𝑢𝑅𝐵) ∧ (𝑢 = 𝐸𝑢𝑅𝐷))))
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 br1cossxrnidres
StepHypRef Expression
1 br1cossxrnres 38429 . 2 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( I ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢 I 𝐶𝑢𝑅𝐵) ∧ (𝑢 I 𝐸𝑢𝑅𝐷))))
2 ideq2 38288 . . . . . 6 (𝑢 ∈ V → (𝑢 I 𝐶𝑢 = 𝐶))
32elv 3482 . . . . 5 (𝑢 I 𝐶𝑢 = 𝐶)
43anbi1i 624 . . . 4 ((𝑢 I 𝐶𝑢𝑅𝐵) ↔ (𝑢 = 𝐶𝑢𝑅𝐵))
5 ideq2 38288 . . . . . 6 (𝑢 ∈ V → (𝑢 I 𝐸𝑢 = 𝐸))
65elv 3482 . . . . 5 (𝑢 I 𝐸𝑢 = 𝐸)
76anbi1i 624 . . . 4 ((𝑢 I 𝐸𝑢𝑅𝐷) ↔ (𝑢 = 𝐸𝑢𝑅𝐷))
84, 7anbi12i 628 . . 3 (((𝑢 I 𝐶𝑢𝑅𝐵) ∧ (𝑢 I 𝐸𝑢𝑅𝐷)) ↔ ((𝑢 = 𝐶𝑢𝑅𝐵) ∧ (𝑢 = 𝐸𝑢𝑅𝐷)))
98rexbii 3091 . 2 (∃𝑢𝐴 ((𝑢 I 𝐶𝑢𝑅𝐵) ∧ (𝑢 I 𝐸𝑢𝑅𝐷)) ↔ ∃𝑢𝐴 ((𝑢 = 𝐶𝑢𝑅𝐵) ∧ (𝑢 = 𝐸𝑢𝑅𝐷)))
101, 9bitrdi 287 1 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( I ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢 = 𝐶𝑢𝑅𝐵) ∧ (𝑢 = 𝐸𝑢𝑅𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wrex 3067  Vcvv 3477  cop 4636   class class class wbr 5147   I cid 5581  cres 5690  cxrn 38160  ccoss 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-1st 8012  df-2nd 8013  df-xrn 38352  df-coss 38392
This theorem is referenced by: (None)
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