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Theorem opeldm 5464
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 4540 . . . 4 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
32eleq1d 2835 . . 3 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
41, 3spcev 3451 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶)
5 opeldm.1 . . 3 𝐴 ∈ V
65eldm2 5458 . 2 (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶)
74, 6sylibr 224 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  cop 4322  dom cdm 5249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-dm 5259
This theorem is referenced by:  breldm  5465  elreldm  5486  relssres  5576  iss  5586  imadmrn  5615  dfco2a  5777  funssres  6071  funun  6073  tz7.48-1  7691  iiner  7971  r0weon  9035  axdc3lem2  9475  uzrdgfni  12961  imasaddfnlem  16392  imasvscafn  16401  cicsym  16667  gsum2d  18574  cffldtocusgr  26574  dfcnv2  29812  bnj1379  31235  cnfin0  33573  cnfinltrel  33574  iss2  34450  rfovcnvf1od  38821
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