Theorem br1cossincnvepres 38406

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 38403	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶))))
2		brcnvep 38221	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐵 ↔ 𝐵 ∈ 𝑢))
3	2	elv 3493	. . . . 5 ⊢ (𝑢◡ E 𝐵 ↔ 𝐵 ∈ 𝑢)
4	3	anbi1i 623	. . . 4 ⊢ ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ↔ (𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵))
5		brcnvep 38221	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢))
6	5	elv 3493	. . . . 5 ⊢ (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢)
7	6	anbi1i 623	. . . 4 ⊢ ((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶) ↔ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶))
8	4, 7	anbi12i 627	. . 3 ⊢ (((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶)))
9	8	rexbii 3100	. 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶)))
10	1, 9	bitrdi 287	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488   ∩ cin 3975   class class class wbr 5166   E cep 5598  ^◡ccnv 5699   ↾ cres 5702   ≀ ccoss 38135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-eprel 5599  df-xp 5706  df-rel 5707  df-cnv 5708  df-res 5712  df-coss 38367

This theorem is referenced by: (None)