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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 5996 and
brrelex1 5735.
Remark: there are many pairs like bj-opelresdm 36657 / bj-brresdm 36658, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 36657 / brrelex12 5734 or the opelopabg 5544 / brabg 5545 family). They are straightforwardly equivalent by df-br 5153. 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 5153 | . 2 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋)) | |
2 | bj-opelresdm 36657 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) | |
3 | 1, 2 | sylbi 216 | 1 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ⟨cop 4638 class class class wbr 5152 ↾ cres 5684 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-res 5694 |
This theorem is referenced by: bj-idreseq 36674 |
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