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Theorem eqbrrdva 4679
Description: Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
Hypotheses
Ref Expression
eqbrrdva.1  |-  ( ph  ->  A  C_  ( C  X.  D ) )
eqbrrdva.2  |-  ( ph  ->  B  C_  ( C  X.  D ) )
eqbrrdva.3  |-  ( (
ph  /\  x  e.  C  /\  y  e.  D
)  ->  ( x A y  <->  x B
y ) )
Assertion
Ref Expression
eqbrrdva  |-  ( ph  ->  A  =  B )
Distinct variable groups:    x, y, A   
x, B, y    ph, x, y
Allowed substitution hints:    C( x, y)    D( x, y)

Proof of Theorem eqbrrdva
StepHypRef Expression
1 eqbrrdva.1 . . . 4  |-  ( ph  ->  A  C_  ( C  X.  D ) )
2 xpss 4617 . . . 4  |-  ( C  X.  D )  C_  ( _V  X.  _V )
31, 2sstrdi 3079 . . 3  |-  ( ph  ->  A  C_  ( _V  X.  _V ) )
4 df-rel 4516 . . 3  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
53, 4sylibr 133 . 2  |-  ( ph  ->  Rel  A )
6 eqbrrdva.2 . . . 4  |-  ( ph  ->  B  C_  ( C  X.  D ) )
76, 2sstrdi 3079 . . 3  |-  ( ph  ->  B  C_  ( _V  X.  _V ) )
8 df-rel 4516 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
97, 8sylibr 133 . 2  |-  ( ph  ->  Rel  B )
101ssbrd 3941 . . . 4  |-  ( ph  ->  ( x A y  ->  x ( C  X.  D ) y ) )
11 brxp 4540 . . . 4  |-  ( x ( C  X.  D
) y  <->  ( x  e.  C  /\  y  e.  D ) )
1210, 11syl6ib 160 . . 3  |-  ( ph  ->  ( x A y  ->  ( x  e.  C  /\  y  e.  D ) ) )
136ssbrd 3941 . . . 4  |-  ( ph  ->  ( x B y  ->  x ( C  X.  D ) y ) )
1413, 11syl6ib 160 . . 3  |-  ( ph  ->  ( x B y  ->  ( x  e.  C  /\  y  e.  D ) ) )
15 eqbrrdva.3 . . . 4  |-  ( (
ph  /\  x  e.  C  /\  y  e.  D
)  ->  ( x A y  <->  x B
y ) )
16153expib 1169 . . 3  |-  ( ph  ->  ( ( x  e.  C  /\  y  e.  D )  ->  (
x A y  <->  x B
y ) ) )
1712, 14, 16pm5.21ndd 679 . 2  |-  ( ph  ->  ( x A y  <-> 
x B y ) )
185, 9, 17eqbrrdv 4606 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   _Vcvv 2660    C_ wss 3041   class class class wbr 3899    X. cxp 4507   Rel wrel 4514
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516
This theorem is referenced by: (None)
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