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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-brresdm | Structured version Visualization version GIF version | ||
| 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 6004 and
     brrelex1 5738. Remark: there are many pairs like bj-opelresdm 37146 / bj-brresdm 37147, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 37146 / brrelex12 5737 or the opelopabg 5543 / brabg 5544 family). They are straightforwardly equivalent by df-br 5144. 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.) | 
| Ref | Expression | 
|---|---|
| bj-brresdm | ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-br 5144 | . 2 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋)) | |
| 2 | bj-opelresdm 37146 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) | |
| 3 | 1, 2 | sylbi 217 | 1 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 ↾ cres 5687 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-res 5697 | 
| This theorem is referenced by: bj-idreseq 37163 | 
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