Theorem br1cossxrnidres 38850

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 38847	. 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 38622	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐶 ↔ 𝑢 = 𝐶))
3	2	elv 3432	. . . . 5 ⊢ (𝑢 I 𝐶 ↔ 𝑢 = 𝐶)
4	3	anbi1i 625	. . . 4 ⊢ ((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ↔ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐵))
5		ideq2 38622	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐸 ↔ 𝑢 = 𝐸))
6	5	elv 3432	. . . . 5 ⊢ (𝑢 I 𝐸 ↔ 𝑢 = 𝐸)
7	6	anbi1i 625	. . . 4 ⊢ ((𝑢 I 𝐸 ∧ 𝑢𝑅𝐷) ↔ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷))
8	4, 7	anbi12i 629	. . 3 ⊢ (((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷)))
9	8	rexbii 3082	. 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 1542   ∈ wcel 2114  ∃wrex 3059  Vcvv 3427  ⟨cop 4563   class class class wbr 5074   I cid 5514   ↾ cres 5622   ⋉ cxrn 38483   ≀ ccoss 38492

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fo 6493  df-fv 6495  df-1st 7931  df-2nd 7932  df-xrn 38689  df-coss 38810

This theorem is referenced by: (None)