Theorem br1cossincnvepres 38148

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 38145	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶))))
2		brcnvep 37963	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐵 ↔ 𝐵 ∈ 𝑢))
3	2	elv 3468	. . . . 5 ⊢ (𝑢◡ E 𝐵 ↔ 𝐵 ∈ 𝑢)
4	3	anbi1i 622	. . . 4 ⊢ ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ↔ (𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵))
5		brcnvep 37963	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢))
6	5	elv 3468	. . . . 5 ⊢ (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢)
7	6	anbi1i 622	. . . 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 3084	. 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 394   ∈ wcel 2099  ∃wrex 3060  Vcvv 3462   ∩ cin 3946   class class class wbr 5153   E cep 5585  ^◡ccnv 5681   ↾ cres 5684   ≀ ccoss 37876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-eprel 5586  df-xp 5688  df-rel 5689  df-cnv 5690  df-res 5694  df-coss 38109

This theorem is referenced by: (None)