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Theorem bj-brresdm 36534
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 5981 and brrelex1 5722.

Remark: there are many pairs like bj-opelresdm 36533 / bj-brresdm 36534, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 36533 / brrelex12 5721 or the opelopabg 5531 / brabg 5532 family). They are straightforwardly equivalent by df-br 5142. 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(𝑅𝑋)𝐵𝐴𝑋)

Proof of Theorem bj-brresdm
StepHypRef Expression
1 df-br 5142 . 2 (𝐴(𝑅𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑋))
2 bj-opelresdm 36533 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑋) → 𝐴𝑋)
31, 2sylbi 216 1 (𝐴(𝑅𝑋)𝐵𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  cop 4629   class class class wbr 5141  cres 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-res 5681
This theorem is referenced by:  bj-idreseq  36550
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