Theorem bj-brresdm 36625

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 5992 and brrelex1 5731.

Remark: there are many pairs like bj-opelresdm 36624 / bj-brresdm 36625, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 36624 / brrelex12 5730 or the opelopabg 5540 / brabg 5541 family). They are straightforwardly equivalent by df-br 5149. 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 5149	. 2 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋))
2		bj-opelresdm 36624	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)
3	1, 2	sylbi 216	1 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   ∈ wcel 2099  ⟨cop 4635   class class class wbr 5148   ↾ cres 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-res 5690

This theorem is referenced by:  bj-idreseq  36641