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Theorem opeldm 5921
Description: Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)
Hypotheses
Ref Expression
opeldm.1 𝐴 ∈ V
opeldm.2 𝐵 ∈ V
Assertion
Ref Expression
opeldm (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶)

Proof of Theorem opeldm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 opeldm.2 . . 3 𝐵 ∈ V
2 opeq2 4879 . . . 4 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
32eleq1d 2824 . . 3 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
41, 3spcev 3606 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶)
5 opeldm.1 . . 3 𝐴 ∈ V
65eldm2 5915 . 2 (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶)
74, 6sylibr 234 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cop 4637  dom cdm 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-dm 5699
This theorem is referenced by:  breldm  5922  elreldm  5949  relssres  6042  iss  6055  imadmrn  6090  dfco2a  6268  relssdmrn  6290  funssres  6612  funun  6614  frxp2  8168  frxp3  8175  frrlem8  8317  frrlem10  8319  tz7.48-1  8482  iiner  8828  r0weon  10050  axdc3lem2  10489  uzrdgfni  13996  imasaddfnlem  17575  imasvscafn  17584  cicsym  17852  gsum2d  20005  noseqrdgfn  28327  cffldtocusgr  29479  cffldtocusgrOLD  29480  dfcnv2  32693  gsumfs2d  33041  bnj1379  34823  iss2  38326  rfovcnvf1od  43994
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