Theorem bj-brresdm 35244

Description: If two classes are related by a restricted binary relation, then the first class is an element of the restricting class. See also brres 5887 and brrelex1 5631.

Remark: there are many pairs like bj-opelresdm 35243 / bj-brresdm 35244, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 35243 / brrelex12 5630 or the opelopabg 5444 / brabg 5445 family). They are straightforwardly equivalent by df-br 5071. The latter is indeed a very direct definition, introducing a "shorthand", and barely necessary, were it not for the frequency of the expression 𝐴𝑅𝐵. Therefore, in the spirit of "definitions are here to be used", most theorems, apart from the most elementary ones, should only have the "br" version, not the "opel" one. (Contributed by BJ, 25-Dec-2023.)

Assertion

Ref	Expression
bj-brresdm	⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋)

Proof of Theorem bj-brresdm

Step	Hyp	Ref	Expression
1		df-br 5071	. 2 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋))
2		bj-opelresdm 35243	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)
3	1, 2	sylbi 216	1 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   ∈ wcel 2108  ⟨cop 4564   class class class wbr 5070   ↾ cres 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-res 5592

This theorem is referenced by:  bj-idreseq  35260