Theorem brcoels 38984

Description: 𝐵 and 𝐶 are coelements : a binary relation. (Contributed by Peter Mazsa, 14-Jan-2020.) (Revised by Peter Mazsa, 5-Oct-2021.)

Assertion

Ref	Expression
brcoels	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∼ 𝐴𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢)))

Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐶

Allowed substitution hints:   𝑉(𝑢)   𝑊(𝑢)

Proof of Theorem brcoels

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2849	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝑢 ↔ 𝐵 ∈ 𝑢))
2		eleq1 2849	. . . 4 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
3	1, 2	bi2anan9 647	. . 3 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢)))
4	3	rexbidv 3185	. 2 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢)))
5		dfcoels 38979	. 2 ⊢ ∼ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
6	4, 5	brabga 5501	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∼ 𝐴𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085   class class class wbr 5097   ∼ ccoels 38643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-eprel 5543  df-xp 5649  df-rel 5650  df-cnv 5651  df-res 5655  df-coss 38960  df-coels 38961

This theorem is referenced by:  erimeq2  39222