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Theorem br1cossxrnidres 38452
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 38449 . 2 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( I ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢 I 𝐶𝑢𝑅𝐵) ∧ (𝑢 I 𝐸𝑢𝑅𝐷))))
2 ideq2 38308 . . . . . 6 (𝑢 ∈ V → (𝑢 I 𝐶𝑢 = 𝐶))
32elv 3485 . . . . 5 (𝑢 I 𝐶𝑢 = 𝐶)
43anbi1i 624 . . . 4 ((𝑢 I 𝐶𝑢𝑅𝐵) ↔ (𝑢 = 𝐶𝑢𝑅𝐵))
5 ideq2 38308 . . . . . 6 (𝑢 ∈ V → (𝑢 I 𝐸𝑢 = 𝐸))
65elv 3485 . . . . 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 3094 . 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 1540  wcel 2108  wrex 3070  Vcvv 3480  cop 4632   class class class wbr 5143   I cid 5577  cres 5687  cxrn 38181  ccoss 38182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-xrn 38372  df-coss 38412
This theorem is referenced by: (None)
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