Theorem breldm 4833

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 4006	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		opeldm.1	. . 3 ⊢ 𝐴 ∈ V
3		opeldm.2	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	opeldm 4832	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → 𝐴 ∈ dom 𝑅)
5	1, 4	sylbi 121	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2148  Vcvv 2739  ⟨cop 3597   class class class wbr 4005  dom cdm 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-dm 4638

This theorem is referenced by:  exse2  5004  funcnv3  5280  dff13  5771  reldmtpos  6256  rntpos  6260  dftpos4  6266  tpostpos  6267  iserd  6563  ntrivcvgap  11558