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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 4654 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | opeldm.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | opeldm.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | opeldm 5328 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 → 𝐴 ∈ dom 𝑅) |
5 | 1, 4 | sylbi 207 | 1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 Vcvv 3200 〈cop 4183 class class class wbr 4653 dom cdm 5114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-dm 5124 |
This theorem is referenced by: funcnv3 5959 opabiota 6261 dffv2 6271 dff13 6512 exse2 7105 reldmtpos 7360 rntpos 7365 dftpos4 7371 tpostpos 7372 wfrlem5 7419 iserd 7768 dcomex 9269 axdc2lem 9270 axdclem2 9342 dmrecnq 9790 cotr2g 13715 shftfval 13810 geolim2 14602 geomulcvg 14607 geoisum1c 14611 cvgrat 14615 ntrivcvg 14629 eftlub 14839 eflegeo 14851 rpnnen2lem5 14947 imasleval 16201 psdmrn 17207 psssdm2 17215 ovoliunnul 23275 vitalilem5 23381 dvcj 23713 dvrec 23718 dvef 23743 ftc1cn 23806 aaliou3lem3 24099 ulmdv 24157 dvradcnv 24175 abelthlem7 24192 abelthlem9 24194 logtayllem 24405 leibpi 24669 log2tlbnd 24672 zetacvg 24741 hhcms 28060 hhsscms 28136 occl 28163 gsummpt2co 29780 iprodgam 31628 imaindm 31682 frrlem5 31784 imageval 32037 knoppcnlem6 32488 knoppndvlem6 32508 knoppf 32526 unccur 33392 ftc1cnnc 33484 geomcau 33555 dvradcnv2 38546 |
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