Theorem bj-opelrelex 37446

Description: The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5672, which could be proved from it. (Contributed by BJ, 27-Dec-2023.)

Assertion

Ref	Expression
bj-opelrelex	⊢ ((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem bj-opelrelex

Step	Hyp	Ref	Expression
1		df-rel 5627	. . . 4 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 216	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	sselda 3917	. 2 ⊢ ((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
4		opelxp 5656	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	3, 4	sylib 218	1 ⊢ ((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3427   ⊆ wss 3885  ⟨cop 4563   × cxp 5618  Rel wrel 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5137  df-xp 5626  df-rel 5627

This theorem is referenced by: (None)