Theorem br1cossxrnidres 37955

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 37952	. 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 37811	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐶 ↔ 𝑢 = 𝐶))
3	2	elv 3479	. . . . 5 ⊢ (𝑢 I 𝐶 ↔ 𝑢 = 𝐶)
4	3	anbi1i 622	. . . 4 ⊢ ((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ↔ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐵))
5		ideq2 37811	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐸 ↔ 𝑢 = 𝐸))
6	5	elv 3479	. . . . 5 ⊢ (𝑢 I 𝐸 ↔ 𝑢 = 𝐸)
7	6	anbi1i 622	. . . 4 ⊢ ((𝑢 I 𝐸 ∧ 𝑢𝑅𝐷) ↔ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷))
8	4, 7	anbi12i 626	. . 3 ⊢ (((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷)))
9	8	rexbii 3091	. 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷)))
10	1, 9	bitrdi 286	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 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3067  Vcvv 3473  ⟨cop 4638   class class class wbr 5152   I cid 5579   ↾ cres 5684   ⋉ cxrn 37680   ≀ ccoss 37681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-1st 7999  df-2nd 8000  df-xrn 37875  df-coss 37915

This theorem is referenced by: (None)