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

Theorem relopabi 4802
Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)
Hypothesis
Ref Expression
relopabi.1 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
relopabi Rel 𝐴

Proof of Theorem relopabi
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 relopabi.1 . . . 4 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 df-opab 4105 . . . 4 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31, 2eqtri 2225 . . 3 𝐴 = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4 vex 2774 . . . . . . . 8 𝑥 ∈ V
5 vex 2774 . . . . . . . 8 𝑦 ∈ V
64, 5opelvv 4724 . . . . . . 7 𝑥, 𝑦⟩ ∈ (V × V)
7 eleq1 2267 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (V × V) ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V)))
86, 7mpbiri 168 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 ∈ (V × V))
98adantr 276 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
109exlimivv 1919 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
1110abssi 3267 . . 3 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (V × V)
123, 11eqsstri 3224 . 2 𝐴 ⊆ (V × V)
13 df-rel 4681 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
1412, 13mpbir 146 1 Rel 𝐴
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1372  wex 1514  wcel 2175  {cab 2190  Vcvv 2771  wss 3165  cop 3635  {copab 4103   × cxp 4672  Rel wrel 4679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-opab 4105  df-xp 4680  df-rel 4681
This theorem is referenced by:  relopab  4803  mptrel  4805  reli  4806  rele  4807  relcnv  5059  cotr  5063  relco  5180  reloprab  5992  reldmoprab  6029  eqer  6651  ecopover  6719  ecopoverg  6722  relen  6830  reldom  6831  enq0er  7547  aprcl  8718  aptap  8722  climrel  11562  brstruct  12812
  Copyright terms: Public domain W3C validator