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Theorem bj-brresdm 34466
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 5841 and brrelex1 5586.

Remark: there are many pairs like bj-opelresdm 34465 / bj-brresdm 34466, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 34465 / brrelex12 5585 or the opelopabg 5406 / brabg 5407 family). They are straightforwardly equivalent by df-br 5048. 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 5048 . 2 (𝐴(𝑅𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑋))
2 bj-opelresdm 34465 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑋) → 𝐴𝑋)
31, 2sylbi 220 1 (𝐴(𝑅𝑋)𝐵𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  cop 4554   class class class wbr 5047  cres 5538
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110  df-xp 5542  df-res 5548
This theorem is referenced by:  bj-idreseq  34482
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