Theorem br1cossincnvepres 39135

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 39132	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶))))
2		brcnvep 38865	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐵 ↔ 𝐵 ∈ 𝑢))
3	2	elv 3458	. . . . 5 ⊢ (𝑢◡ E 𝐵 ↔ 𝐵 ∈ 𝑢)
4	3	anbi1i 635	. . . 4 ⊢ ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ↔ (𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵))
5		brcnvep 38865	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢))
6	5	elv 3458	. . . . 5 ⊢ (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢)
7	6	anbi1i 635	. . . 4 ⊢ ((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶) ↔ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶))
8	4, 7	anbi12i 639	. . 3 ⊢ (((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶)))
9	8	rexbii 3110	. 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶)))
10	1, 9	bitrdi 290	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2141  ∃wrex 3087  Vcvv 3453   ∩ cin 3903   class class class wbr 5108   E cep 5560  ^◡ccnv 5660   ↾ cres 5663   ≀ ccoss 38778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-res 5673  df-coss 39096

This theorem is referenced by: (None)