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Theorem opeldm 5748
 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 4764 . . . 4 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
32eleq1d 2837 . . 3 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
41, 3spcev 3526 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶)
5 opeldm.1 . . 3 𝐴 ∈ V
65eldm2 5742 . 2 (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶)
74, 6sylibr 237 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539  ∃wex 1782   ∈ wcel 2112  Vcvv 3410  ⟨cop 4529  dom cdm 5525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-un 3864  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-dm 5535 This theorem is referenced by:  breldm  5749  elreldm  5777  relssres  5865  iss  5876  imadmrn  5912  dfco2a  6077  funssres  6380  funun  6382  tz7.48-1  8090  iiner  8380  r0weon  9473  axdc3lem2  9912  uzrdgfni  13376  imasaddfnlem  16860  imasvscafn  16869  cicsym  17134  gsum2d  19161  cffldtocusgr  27337  dfcnv2  30538  bnj1379  32331  frxp2  33347  frxp3  33353  frrlem8  33392  frrlem10  33394  iss2  36042  rfovcnvf1od  41079
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