Theorem bj-brresdm 37458

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 5943 and brrelex1 5675.

Remark: there are many pairs like bj-opelresdm 37457 / bj-brresdm 37458, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 37457 / brrelex12 5674 or the opelopabg 5484 / brabg 5485 family). They are straightforwardly equivalent by df-br 5087. 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 5087	. 2 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋))
2		bj-opelresdm 37457	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)
3	1, 2	sylbi 217	1 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   ∈ wcel 2114  ⟨cop 4574   class class class wbr 5086   ↾ cres 5624

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 2709  ax-sep 5231  ax-pr 5368

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5628  df-res 5634

This theorem is referenced by:  bj-idreseq  37474