Theorem bj-brresdm 37650

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 5976 and brrelex1 5705.

Remark: there are many pairs like bj-opelresdm 37649 / bj-brresdm 37650, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 37649 / brrelex12 5704 or the opelopabg 5514 / brabg 5515 family). They are straightforwardly equivalent by df-br 5106. 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 5106	. 2 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋))
2		bj-opelresdm 37649	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)
3	1, 2	sylbi 220	1 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   ∈ wcel 2145  ⟨cop 4591   class class class wbr 5105   ↾ cres 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-res 5664

This theorem is referenced by:  bj-idreseq  37666