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Theorem opabid2 4410
Description: A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
Assertion
Ref Expression
opabid2 (Rel A → {⟨x, y⟩ ∣ ⟨x, y A} = A)
Distinct variable group:   x,y,A

Proof of Theorem opabid2
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . 4 z V
2 vex 2554 . . . 4 w V
3 opeq1 3540 . . . . 5 (x = z → ⟨x, y⟩ = ⟨z, y⟩)
43eleq1d 2103 . . . 4 (x = z → (⟨x, y A ↔ ⟨z, y A))
5 opeq2 3541 . . . . 5 (y = w → ⟨z, y⟩ = ⟨z, w⟩)
65eleq1d 2103 . . . 4 (y = w → (⟨z, y A ↔ ⟨z, w A))
71, 2, 4, 6opelopab 3999 . . 3 (⟨z, w {⟨x, y⟩ ∣ ⟨x, y A} ↔ ⟨z, w A)
87gen2 1336 . 2 zw(⟨z, w {⟨x, y⟩ ∣ ⟨x, y A} ↔ ⟨z, w A)
9 relopab 4407 . . 3 Rel {⟨x, y⟩ ∣ ⟨x, y A}
10 eqrel 4372 . . 3 ((Rel {⟨x, y⟩ ∣ ⟨x, y A} Rel A) → ({⟨x, y⟩ ∣ ⟨x, y A} = Azw(⟨z, w {⟨x, y⟩ ∣ ⟨x, y A} ↔ ⟨z, w A)))
119, 10mpan 400 . 2 (Rel A → ({⟨x, y⟩ ∣ ⟨x, y A} = Azw(⟨z, w {⟨x, y⟩ ∣ ⟨x, y A} ↔ ⟨z, w A)))
128, 11mpbiri 157 1 (Rel A → {⟨x, y⟩ ∣ ⟨x, y A} = A)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242   wcel 1390  cop 3370  {copab 3808  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-eu 1900  df-mo 1901  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:  opabbi2dv  4428
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