Theorem breldm 5806

Description: Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)

Hypotheses

Ref	Expression
opeldm.1	⊢ 𝐴 ∈ V
opeldm.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
breldm	⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Proof of Theorem breldm

Step	Hyp	Ref	Expression
1		df-br 5071	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		opeldm.1	. . 3 ⊢ 𝐴 ∈ V
3		opeldm.2	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	opeldm 5805	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → 𝐴 ∈ dom 𝑅)
5	1, 4	sylbi 216	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564   class class class wbr 5070  dom cdm 5580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-dm 5590

This theorem is referenced by:  funcnv3  6488  opabiota  6833  dffv2  6845  dff13  7109  exse2  7738  reldmtpos  8021  rntpos  8026  dftpos4  8032  tpostpos  8033  fprlem1  8087  wfrlem5OLD  8115  iserd  8482  frrlem15  9446  dcomex  10134  axdc2lem  10135  dmrecnq  10655  cotr2g  14615  shftfval  14709  geolim2  15511  geomulcvg  15516  geoisum1c  15520  cvgrat  15523  ntrivcvg  15537  eftlub  15746  eflegeo  15758  rpnnen2lem5  15855  imasleval  17169  psdmrn  18206  psssdm2  18214  ovoliunnul  24576  vitalilem5  24681  dvcj  25019  dvrec  25024  dvef  25049  ftc1cn  25112  aaliou3lem3  25409  ulmdv  25467  dvradcnv  25485  abelthlem7  25502  abelthlem9  25504  logtayllem  25719  leibpi  25997  log2tlbnd  26000  zetacvg  26069  hhcms  29466  hhsscms  29541  occl  29567  gsummpt2co  31210  iprodgam  33614  imaindm  33659  dmttrcl  33707  ttrclse  33713  imageval  34159  knoppcnlem6  34605  knoppndvlem6  34624  knoppf  34642  unccur  35687  ftc1cnnc  35776  geomcau  35844  dvradcnv2  41854