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Theorem eqbrriv 4973
Description: Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
Hypotheses
Ref Expression
eqbrriv.1  |-  Rel  A
eqbrriv.2  |-  Rel  B
eqbrriv.3  |-  ( x A y  <->  x B
y )
Assertion
Ref Expression
eqbrriv  |-  A  =  B
Distinct variable groups:    x, y, A    x, B, y

Proof of Theorem eqbrriv
StepHypRef Expression
1 eqbrriv.1 . 2  |-  Rel  A
2 eqbrriv.2 . 2  |-  Rel  B
3 eqbrriv.3 . . 3  |-  ( x A y  <->  x B
y )
4 df-br 4215 . . 3  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
5 df-br 4215 . . 3  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
63, 4, 53bitr3i 268 . 2  |-  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )
71, 2, 6eqrelriiv 4972 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   <.cop 3819   class class class wbr 4214   Rel wrel 4885
This theorem is referenced by:  resco  5376  tpostpos  6501  sbthcl  7231  dfle2  10742  dflt2  10743  idsset  25737  dfbigcup2  25746  imageval  25777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887
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