Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  opeldm GIF version

Theorem opeldm 4742
 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 3706 . . . 4 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
32eleq1d 2208 . . 3 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
41, 3spcev 2780 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶)
5 opeldm.1 . . 3 𝐴 ∈ V
65eldm2 4737 . 2 (𝐴 ∈ dom 𝐶 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐶)
74, 6sylibr 133 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686  ⟨cop 3530  dom cdm 4539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-dm 4549 This theorem is referenced by:  breldm  4743  elreldm  4765  relssres  4857  iss  4865  imadmrn  4891  dfco2a  5039  funssres  5165  funun  5167  iinerm  6501
 Copyright terms: Public domain W3C validator