Theorem elrelb 5774

Description: A member of a relation expressed by an ordered pair. (Contributed by AV, 23-Aug-2026.)

Assertion

Ref	Expression
elrelb	⊢ (Rel 𝑅 → (𝐴 ∈ 𝑅 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem elrelb

Step	Hyp	Ref	Expression
1		elrel 5773	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		eleq1 2848	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
3		df-br 5103	. . . . . . . . . 10 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
4	3	biimpri 231	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → 𝑥𝑅𝑦)
5	2, 4	biimtrdi 256	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 → 𝑥𝑅𝑦))
6	5	com12 33	. . . . . . 7 ⊢ (𝐴 ∈ 𝑅 → (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥𝑅𝑦))
7	6	adantl 487	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥𝑅𝑦))
8	7	ancld 560	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)))
9	8	2eximdv 1951	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)))
10	1, 9	mpd 16	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
11	10	ex 418	. 2 ⊢ (Rel 𝑅 → (𝐴 ∈ 𝑅 → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)))
12	3	bilani 510	. . . . 5 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → ⟨𝑥, 𝑦⟩ ∈ 𝑅)
13	2	adantr 486	. . . . 5 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → (𝐴 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
14	12, 13	mpbird 260	. . . 4 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅)
15	14	exlimiv 1962	. . 3 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅)
16	15	exlimiv 1962	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅)
17	11, 16	impbid1 228	1 ⊢ (Rel 𝑅 → (𝐴 ∈ 𝑅 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ⟨cop 4589   class class class wbr 5102  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-in 4091  df-un 4151  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657

This theorem is used by:  dfric2  20757