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Mirrors > Home > ILE Home > Th. List > df-br | GIF version |
Description: Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. This definition of relations is well-defined, although not very meaningful, when classes 𝐴 and/or 𝐵 are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when 𝑅 is a proper class (see for example iprc 4895). (Contributed by NM, 31-Dec-1993.) |
Ref | Expression |
---|---|
df-br | ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cB | . . 3 class 𝐵 | |
3 | cR | . . 3 class 𝑅 | |
4 | 1, 2, 3 | wbr 4003 | . 2 wff 𝐴𝑅𝐵 |
5 | 1, 2 | cop 3595 | . . 3 class ⟨𝐴, 𝐵⟩ |
6 | 5, 3 | wcel 2148 | . 2 wff ⟨𝐴, 𝐵⟩ ∈ 𝑅 |
7 | 4, 6 | wb 105 | 1 wff (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅) |
Colors of variables: wff set class |
This definition is referenced by: breq 4005 breq1 4006 breq2 4007 ssbrd 4046 nfbrd 4048 br0 4051 brne0 4052 brm 4053 brun 4054 brin 4055 brdif 4056 opabss 4067 brabsb 4261 brabga 4264 epelg 4290 pofun 4312 brxp 4657 brab2a 4679 brab2ga 4701 eqbrriv 4721 eqbrrdv 4723 eqbrrdiv 4724 opeliunxp2 4767 opelco2g 4795 opelco 4799 cnvss 4800 elcnv2 4805 opelcnvg 4807 brcnvg 4808 dfdm3 4814 dfrn3 4816 elrng 4818 eldm2g 4823 breldm 4831 dmopab 4838 opelrng 4859 opelrn 4861 elrn 4870 rnopab 4874 brres 4913 brresg 4915 resieq 4917 opelresi 4918 resiexg 4952 iss 4953 dfres2 4959 restidsing 4963 dfima3 4973 elima3 4977 imai 4984 elimasn 4995 eliniseg 4998 cotr 5010 issref 5011 cnvsym 5012 intasym 5013 asymref 5014 intirr 5015 codir 5017 qfto 5018 poirr2 5021 dmsnm 5094 coiun 5138 co02 5142 coi1 5144 dffun4 5227 dffun4f 5232 funeu2 5242 funopab 5251 funco 5256 funcnvsn 5261 isarep1 5302 fnop 5319 fneu2 5321 brprcneu 5508 dffv3g 5511 tz6.12 5543 nfvres 5548 0fv 5550 funopfv 5555 fnopfvb 5557 fvmptss2 5591 funfvbrb 5629 dff3im 5661 dff4im 5662 f1ompt 5667 idref 5757 foeqcnvco 5790 f1eqcocnv 5791 fliftel 5793 fliftel1 5794 fliftcnv 5795 f1oiso 5826 ovprc 5909 brabvv 5920 1st2ndbr 6184 xporderlem 6231 opeliunxp2f 6238 rbropapd 6242 ottposg 6255 dftpos3 6262 dftpos4 6263 tposoprab 6280 tfrlem7 6317 tfrexlem 6334 ercnv 6555 brdifun 6561 swoord1 6563 swoord2 6564 0er 6568 elecg 6572 iinerm 6606 brecop 6624 idssen 6776 xpcomco 6825 netap 7252 2omotaplemap 7255 exmidapne 7258 ltdfpr 7504 xrlenlt 8021 aprcl 8602 frecuzrdgtcl 10411 frecuzrdgfunlem 10418 climcau 11354 divides 11795 isstructim 12475 isstructr 12476 imasaddfnlemg 12734 subrgdvds 13354 aprval 13370 lmrcl 13661 lmff 13719 xmeterval 13905 eldvap 14121 dvef 14158 |
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