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Theorem opeldm 5907
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 4874 . . . 4 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
32eleq1d 2818 . . 3 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
41, 3spcev 3596 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶)
5 opeldm.1 . . 3 𝐴 ∈ V
65eldm2 5901 . 2 (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶)
74, 6sylibr 233 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wex 1781  wcel 2106  Vcvv 3474  cop 4634  dom cdm 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-dm 5686
This theorem is referenced by:  breldm  5908  elreldm  5934  relssres  6022  iss  6035  imadmrn  6069  dfco2a  6245  relssdmrn  6267  funssres  6592  funun  6594  frxp2  8129  frxp3  8136  frrlem8  8277  frrlem10  8279  tz7.48-1  8442  iiner  8782  r0weon  10006  axdc3lem2  10445  uzrdgfni  13922  imasaddfnlem  17473  imasvscafn  17482  cicsym  17750  gsum2d  19839  cffldtocusgr  28701  dfcnv2  31896  bnj1379  33836  iss2  37208  rfovcnvf1od  42745
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