Theorem br1cossincnvepres 36495

Description: 𝐵 and 𝐶 are cosets by an intersection with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)

Assertion

Ref	Expression
br1cossincnvepres	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶))))

Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐶   𝑢,𝑅   𝑢,𝑉   𝑢,𝑊

Proof of Theorem br1cossincnvepres

Step	Hyp	Ref	Expression
1		br1cossinres 36492	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶))))
2		brcnvep 36331	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐵 ↔ 𝐵 ∈ 𝑢))
3	2	elv 3428	. . . . 5 ⊢ (𝑢◡ E 𝐵 ↔ 𝐵 ∈ 𝑢)
4	3	anbi1i 623	. . . 4 ⊢ ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ↔ (𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵))
5		brcnvep 36331	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢))
6	5	elv 3428	. . . . 5 ⊢ (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢)
7	6	anbi1i 623	. . . 4 ⊢ ((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶) ↔ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶))
8	4, 7	anbi12i 626	. . 3 ⊢ (((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶)))
9	8	rexbii 3177	. 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶)))
10	1, 9	bitrdi 286	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   class class class wbr 5070   E cep 5485  ^◡ccnv 5579   ↾ cres 5582   ≀ ccoss 36260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-eprel 5486  df-xp 5586  df-rel 5587  df-cnv 5588  df-res 5592  df-coss 36464

This theorem is referenced by: (None)