ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqbrrdva Unicode version

Theorem eqbrrdva 4594
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 4534 . . . 4  |-  ( C  X.  D )  C_  ( _V  X.  _V )
31, 2syl6ss 3035 . . 3  |-  ( ph  ->  A  C_  ( _V  X.  _V ) )
4 df-rel 4435 . . 3  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
53, 4sylibr 132 . 2  |-  ( ph  ->  Rel  A )
6 eqbrrdva.2 . . . 4  |-  ( ph  ->  B  C_  ( C  X.  D ) )
76, 2syl6ss 3035 . . 3  |-  ( ph  ->  B  C_  ( _V  X.  _V ) )
8 df-rel 4435 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
97, 8sylibr 132 . 2  |-  ( ph  ->  Rel  B )
101ssbrd 3878 . . . 4  |-  ( ph  ->  ( x A y  ->  x ( C  X.  D ) y ) )
11 brxp 4458 . . . 4  |-  ( x ( C  X.  D
) y  <->  ( x  e.  C  /\  y  e.  D ) )
1210, 11syl6ib 159 . . 3  |-  ( ph  ->  ( x A y  ->  ( x  e.  C  /\  y  e.  D ) ) )
136ssbrd 3878 . . . 4  |-  ( ph  ->  ( x B y  ->  x ( C  X.  D ) y ) )
1413, 11syl6ib 159 . . 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 1146 . . 3  |-  ( ph  ->  ( ( x  e.  C  /\  y  e.  D )  ->  (
x A y  <->  x B
y ) ) )
1712, 14, 16pm5.21ndd 656 . 2  |-  ( ph  ->  ( x A y  <-> 
x B y ) )
185, 9, 17eqbrrdv 4523 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   _Vcvv 2619    C_ wss 2997   class class class wbr 3837    X. cxp 4426   Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator