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Mirrors > Home > MPE Home > Th. List > breldm | Structured version Visualization version GIF version |
Description: Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.) |
Ref | Expression |
---|---|
opeldm.1 | ⊢ 𝐴 ∈ V |
opeldm.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
breldm | ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5148 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | opeldm.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | opeldm.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | opeldm 5920 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 → 𝐴 ∈ dom 𝑅) |
5 | 1, 4 | sylbi 217 | 1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3477 〈cop 4636 class class class wbr 5147 dom cdm 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-dm 5698 |
This theorem is referenced by: imaindm 6320 funcnv3 6637 opabiota 6990 dffv2 7003 dff13 7274 exse2 7939 reldmtpos 8257 rntpos 8262 dftpos4 8268 tpostpos 8269 fprlem1 8323 wfrlem5OLD 8351 iserd 8769 dmttrcl 9758 ttrclse 9764 frrlem15 9794 dcomex 10484 axdc2lem 10485 dmrecnq 11005 cotr2g 15011 shftfval 15105 geolim2 15903 geomulcvg 15908 geoisum1c 15912 cvgrat 15915 ntrivcvg 15929 eftlub 16141 eflegeo 16153 rpnnen2lem5 16250 imasleval 17587 psdmrn 18630 psssdm2 18638 ovoliunnul 25555 vitalilem5 25660 dvcj 26002 dvrec 26007 dvef 26032 ftc1cn 26098 aaliou3lem3 26400 ulmdv 26460 dvradcnv 26478 abelthlem7 26496 abelthlem9 26498 logtayllem 26715 leibpi 26999 log2tlbnd 27002 zetacvg 27072 hhcms 31231 hhsscms 31306 occl 31332 gsummpt2co 33033 iprodgam 35721 imageval 35911 knoppcnlem6 36480 knoppndvlem6 36499 knoppf 36517 unccur 37589 ftc1cnnc 37678 geomcau 37745 dvradcnv2 44342 |
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