Theorem brcoels 39234

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 2853	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝑢 ↔ 𝐵 ∈ 𝑢))
2		eleq1 2853	. . . 4 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
3	1, 2	bi2anan9 650	. . 3 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢)))
4	3	rexbidv 3191	. 2 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢)))
5		dfcoels 39229	. 2 ⊢ ∼ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
6	4, 5	brabga 5520	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∼ 𝐴𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   class class class wbr 5111   ∼ ccoels 38893

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-eprel 5563  df-xp 5669  df-rel 5670  df-cnv 5671  df-res 5675  df-coss 39210  df-coels 39211

This theorem is used by:  erimeq2  39472