Theorem bj-brresdm 38035

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 5977 and brrelex1 5704.

Remark: there are many pairs like bj-opelresdm 38034 / bj-brresdm 38035, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 38034 / brrelex12 5703 or the opelopabg 5513 / brabg 5514 family). They are straightforwardly equivalent by df-br 5104. 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 5104	. 2 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋))
2		bj-opelresdm 38034	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)
3	1, 2	sylbi 220	1 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ⟨cop 4590   class class class wbr 5103   ↾ cres 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-res 5663

This theorem is used by:  bj-idreseq  38051