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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 5853 and
brrelex1 5598.
Remark: there are many pairs like bj-opelresdm 34459 / bj-brresdm 34460, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 34459 / brrelex12 5597 or the opelopabg 5418 / brabg 5419 family). They are straightforwardly equivalent by df-br 5060. 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 5060 | . 2 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋)) | |
2 | bj-opelresdm 34459 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) | |
3 | 1, 2 | sylbi 219 | 1 ⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 〈cop 4566 class class class wbr 5059 ↾ cres 5550 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-xp 5554 df-res 5560 |
This theorem is referenced by: bj-idreseq 34476 |
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