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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 5825 and
brrelex1 5569.
Remark: there are many pairs like bj-opelresdm 34560 / bj-brresdm 34561, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 34560 / brrelex12 5568 or the opelopabg 5390 / brabg 5391 family). They are straightforwardly equivalent by df-br 5031. 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 5031 | . 2 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋)) | |
2 | bj-opelresdm 34560 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) | |
3 | 1, 2 | sylbi 220 | 1 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 ↾ cres 5521 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
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 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-res 5531 |
This theorem is referenced by: bj-idreseq 34577 |
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