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Theorem eqbrrdva 5014
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 4951 . . . 4  |-  ( C  X.  D )  C_  ( _V  X.  _V )
31, 2syl6ss 3373 . . 3  |-  ( ph  ->  A  C_  ( _V  X.  _V ) )
4 df-rel 4852 . . 3  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
53, 4sylibr 212 . 2  |-  ( ph  ->  Rel  A )
6 eqbrrdva.2 . . . 4  |-  ( ph  ->  B  C_  ( C  X.  D ) )
76, 2syl6ss 3373 . . 3  |-  ( ph  ->  B  C_  ( _V  X.  _V ) )
8 df-rel 4852 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
97, 8sylibr 212 . 2  |-  ( ph  ->  Rel  B )
101ssbrd 4338 . . . 4  |-  ( ph  ->  ( x A y  ->  x ( C  X.  D ) y ) )
11 brxp 4875 . . . 4  |-  ( x ( C  X.  D
) y  <->  ( x  e.  C  /\  y  e.  D ) )
1210, 11syl6ib 226 . . 3  |-  ( ph  ->  ( x A y  ->  ( x  e.  C  /\  y  e.  D ) ) )
136ssbrd 4338 . . . 4  |-  ( ph  ->  ( x B y  ->  x ( C  X.  D ) y ) )
1413, 11syl6ib 226 . . 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 1190 . . 3  |-  ( ph  ->  ( ( x  e.  C  /\  y  e.  D )  ->  (
x A y  <->  x B
y ) ) )
1712, 14, 16pm5.21ndd 354 . 2  |-  ( ph  ->  ( x A y  <-> 
x B y ) )
185, 9, 17eqbrrdv 4942 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2977    C_ wss 3333   class class class wbr 4297    X. cxp 4843   Rel wrel 4850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852
This theorem is referenced by:  metustsymOLD  20141  metustsym  20142
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