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Theorem eqbrrdva 5015
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 4952 . . . 4  |-  ( C  X.  D )  C_  ( _V  X.  _V )
31, 2syl6ss 3473 . . 3  |-  ( ph  ->  A  C_  ( _V  X.  _V ) )
4 df-rel 4852 . . 3  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
53, 4sylibr 215 . 2  |-  ( ph  ->  Rel  A )
6 eqbrrdva.2 . . . 4  |-  ( ph  ->  B  C_  ( C  X.  D ) )
76, 2syl6ss 3473 . . 3  |-  ( ph  ->  B  C_  ( _V  X.  _V ) )
8 df-rel 4852 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
97, 8sylibr 215 . 2  |-  ( ph  ->  Rel  B )
101ssbrd 4458 . . . 4  |-  ( ph  ->  ( x A y  ->  x ( C  X.  D ) y ) )
11 brxp 4876 . . . 4  |-  ( x ( C  X.  D
) y  <->  ( x  e.  C  /\  y  e.  D ) )
1210, 11syl6ib 229 . . 3  |-  ( ph  ->  ( x A y  ->  ( x  e.  C  /\  y  e.  D ) ) )
136ssbrd 4458 . . . 4  |-  ( ph  ->  ( x B y  ->  x ( C  X.  D ) y ) )
1413, 11syl6ib 229 . . 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 1208 . . 3  |-  ( ph  ->  ( ( x  e.  C  /\  y  e.  D )  ->  (
x A y  <->  x B
y ) ) )
1712, 14, 16pm5.21ndd 355 . 2  |-  ( ph  ->  ( x A y  <-> 
x B y ) )
185, 9, 17eqbrrdv 4943 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   _Vcvv 3078    C_ wss 3433   class class class wbr 4417    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 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-rel 4852
This theorem is referenced by:  metustsym  21494
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