Theorem brcoels 38433

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 2817	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝑢 ↔ 𝐵 ∈ 𝑢))
2		eleq1 2817	. . . 4 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
3	1, 2	bi2anan9 638	. . 3 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢)))
4	3	rexbidv 3158	. 2 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢)))
5		dfcoels 38428	. 2 ⊢ ∼ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
6	4, 5	brabga 5497	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∼ 𝐴𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3054   class class class wbr 5110   ∼ ccoels 38177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-eprel 5541  df-xp 5647  df-rel 5648  df-cnv 5649  df-res 5653  df-coss 38409  df-coels 38410

This theorem is referenced by:  erimeq2  38677