Theorem br1cossxrnidres 38438

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	⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( I ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷))))

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

Step	Hyp	Ref	Expression
1		br1cossxrnres 38435	. 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( I ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐸 ∧ 𝑢𝑅𝐷))))
2		ideq2 38291	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐶 ↔ 𝑢 = 𝐶))
3	2	elv 3441	. . . . 5 ⊢ (𝑢 I 𝐶 ↔ 𝑢 = 𝐶)
4	3	anbi1i 624	. . . 4 ⊢ ((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ↔ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐵))
5		ideq2 38291	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐸 ↔ 𝑢 = 𝐸))
6	5	elv 3441	. . . . 5 ⊢ (𝑢 I 𝐸 ↔ 𝑢 = 𝐸)
7	6	anbi1i 624	. . . 4 ⊢ ((𝑢 I 𝐸 ∧ 𝑢𝑅𝐷) ↔ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷))
8	4, 7	anbi12i 628	. . 3 ⊢ (((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷)))
9	8	rexbii 3076	. 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷)))
10	1, 9	bitrdi 287	1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( I ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3436  ⟨cop 4583   class class class wbr 5092   I cid 5513   ↾ cres 5621   ⋉ cxrn 38164   ≀ ccoss 38165

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fo 6488  df-fv 6490  df-1st 7924  df-2nd 7925  df-xrn 38349  df-coss 38398

This theorem is referenced by: (None)