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Theorem relopabi 4406
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 3810 . . . 4 {⟨x, y⟩ ∣ φ} = {zxy(z = ⟨x, y φ)}
31, 2eqtri 2057 . . 3 A = {zxy(z = ⟨x, y φ)}
4 vex 2554 . . . . . . . 8 x V
5 vex 2554 . . . . . . . 8 y V
64, 5opelvv 4333 . . . . . . 7 x, y (V × V)
7 eleq1 2097 . . . . . . 7 (z = ⟨x, y⟩ → (z (V × V) ↔ ⟨x, y (V × V)))
86, 7mpbiri 157 . . . . . 6 (z = ⟨x, y⟩ → z (V × V))
98adantr 261 . . . . 5 ((z = ⟨x, y φ) → z (V × V))
109exlimivv 1773 . . . 4 (xy(z = ⟨x, y φ) → z (V × V))
1110abssi 3009 . . 3 {zxy(z = ⟨x, y φ)} ⊆ (V × V)
123, 11eqsstri 2969 . 2 A ⊆ (V × V)
13 df-rel 4295 . 2 (Rel AA ⊆ (V × V))
1412, 13mpbir 134 1 Rel A
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  Vcvv 2551  wss 2911  cop 3370  {copab 3808   × cxp 4286  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294  df-rel 4295
This theorem is referenced by:  relopab  4407  reli  4408  rele  4409  relcnv  4646  cotr  4649  relco  4762  reloprab  5495  reldmoprab  5531  eqer  6074  ecopover  6140  ecopoverg  6143  relen  6161  reldom  6162  enq0er  6417
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