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Theorem eqbrrdva 4533
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 4474 . . . 4  |-  ( C  X.  D )  C_  ( _V  X.  _V )
31, 2syl6ss 2985 . . 3  |-  ( ph  ->  A  C_  ( _V  X.  _V ) )
4 df-rel 4380 . . 3  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
53, 4sylibr 141 . 2  |-  ( ph  ->  Rel  A )
6 eqbrrdva.2 . . . 4  |-  ( ph  ->  B  C_  ( C  X.  D ) )
76, 2syl6ss 2985 . . 3  |-  ( ph  ->  B  C_  ( _V  X.  _V ) )
8 df-rel 4380 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
97, 8sylibr 141 . 2  |-  ( ph  ->  Rel  B )
101ssbrd 3833 . . . 4  |-  ( ph  ->  ( x A y  ->  x ( C  X.  D ) y ) )
11 brxp 4403 . . . 4  |-  ( x ( C  X.  D
) y  <->  ( x  e.  C  /\  y  e.  D ) )
1210, 11syl6ib 154 . . 3  |-  ( ph  ->  ( x A y  ->  ( x  e.  C  /\  y  e.  D ) ) )
136ssbrd 3833 . . . 4  |-  ( ph  ->  ( x B y  ->  x ( C  X.  D ) y ) )
1413, 11syl6ib 154 . . 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 1118 . . 3  |-  ( ph  ->  ( ( x  e.  C  /\  y  e.  D )  ->  (
x A y  <->  x B
y ) ) )
1712, 14, 16pm5.21ndd 631 . 2  |-  ( ph  ->  ( x A y  <-> 
x B y ) )
185, 9, 17eqbrrdv 4465 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   _Vcvv 2574    C_ wss 2945   class class class wbr 3792    X. cxp 4371   Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380
This theorem is referenced by: (None)
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