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Theorem relopabi 4633
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 3958 . . . 4 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31, 2eqtri 2136 . . 3 𝐴 = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4 vex 2661 . . . . . . . 8 𝑥 ∈ V
5 vex 2661 . . . . . . . 8 𝑦 ∈ V
64, 5opelvv 4557 . . . . . . 7 𝑥, 𝑦⟩ ∈ (V × V)
7 eleq1 2178 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (V × V) ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V)))
86, 7mpbiri 167 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 ∈ (V × V))
98adantr 272 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
109exlimivv 1850 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
1110abssi 3140 . . 3 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (V × V)
123, 11eqsstri 3097 . 2 𝐴 ⊆ (V × V)
13 df-rel 4514 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
1412, 13mpbir 145 1 Rel 𝐴
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wex 1451  wcel 1463  {cab 2101  Vcvv 2658  wss 3039  cop 3498  {copab 3956   × cxp 4505  Rel wrel 4512
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-xp 4513  df-rel 4514
This theorem is referenced by:  relopab  4634  mptrel  4635  reli  4636  rele  4637  relcnv  4885  cotr  4888  relco  5005  reloprab  5785  reldmoprab  5822  eqer  6427  ecopover  6493  ecopoverg  6496  relen  6604  reldom  6605  enq0er  7207  aprcl  8370  climrel  10989  brstruct  11863
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