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Theorem elreldm 4649
Description: The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
Assertion
Ref Expression
elreldm  |-  ( ( Rel  A  /\  B  e.  A )  ->  |^| |^| B  e.  dom  A )

Proof of Theorem elreldm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rel 4435 . . . . 5  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
2 ssel 3017 . . . . 5  |-  ( A 
C_  ( _V  X.  _V )  ->  ( B  e.  A  ->  B  e.  ( _V  X.  _V ) ) )
31, 2sylbi 119 . . . 4  |-  ( Rel 
A  ->  ( B  e.  A  ->  B  e.  ( _V  X.  _V ) ) )
4 elvv 4488 . . . 4  |-  ( B  e.  ( _V  X.  _V )  <->  E. x E. y  B  =  <. x ,  y >. )
53, 4syl6ib 159 . . 3  |-  ( Rel 
A  ->  ( B  e.  A  ->  E. x E. y  B  =  <. x ,  y >.
) )
6 eleq1 2150 . . . . . 6  |-  ( B  =  <. x ,  y
>.  ->  ( B  e.  A  <->  <. x ,  y
>.  e.  A ) )
7 vex 2622 . . . . . . 7  |-  x  e. 
_V
8 vex 2622 . . . . . . 7  |-  y  e. 
_V
97, 8opeldm 4627 . . . . . 6  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
106, 9syl6bi 161 . . . . 5  |-  ( B  =  <. x ,  y
>.  ->  ( B  e.  A  ->  x  e.  dom  A ) )
11 inteq 3686 . . . . . . . 8  |-  ( B  =  <. x ,  y
>.  ->  |^| B  =  |^| <.
x ,  y >.
)
1211inteqd 3688 . . . . . . 7  |-  ( B  =  <. x ,  y
>.  ->  |^| |^| B  =  |^| |^|
<. x ,  y >.
)
137, 8op1stb 4290 . . . . . . 7  |-  |^| |^| <. x ,  y >.  =  x
1412, 13syl6eq 2136 . . . . . 6  |-  ( B  =  <. x ,  y
>.  ->  |^| |^| B  =  x )
1514eleq1d 2156 . . . . 5  |-  ( B  =  <. x ,  y
>.  ->  ( |^| |^| B  e.  dom  A  <->  x  e.  dom  A ) )
1610, 15sylibrd 167 . . . 4  |-  ( B  =  <. x ,  y
>.  ->  ( B  e.  A  ->  |^| |^| B  e.  dom  A ) )
1716exlimivv 1824 . . 3  |-  ( E. x E. y  B  =  <. x ,  y
>.  ->  ( B  e.  A  ->  |^| |^| B  e.  dom  A ) )
185, 17syli 37 . 2  |-  ( Rel 
A  ->  ( B  e.  A  ->  |^| |^| B  e.  dom  A ) )
1918imp 122 1  |-  ( ( Rel  A  /\  B  e.  A )  ->  |^| |^| B  e.  dom  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619    C_ wss 2997   <.cop 3444   |^|cint 3683    X. cxp 4426   dom cdm 4428   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-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-int 3684  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-dm 4438
This theorem is referenced by:  1stdm  5934  fundmen  6503
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