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Theorem elreldm 4588
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 4380 . . . . 5  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
2 ssel 2967 . . . . 5  |-  ( A 
C_  ( _V  X.  _V )  ->  ( B  e.  A  ->  B  e.  ( _V  X.  _V ) ) )
31, 2sylbi 118 . . . 4  |-  ( Rel 
A  ->  ( B  e.  A  ->  B  e.  ( _V  X.  _V ) ) )
4 elvv 4430 . . . 4  |-  ( B  e.  ( _V  X.  _V )  <->  E. x E. y  B  =  <. x ,  y >. )
53, 4syl6ib 154 . . 3  |-  ( Rel 
A  ->  ( B  e.  A  ->  E. x E. y  B  =  <. x ,  y >.
) )
6 eleq1 2116 . . . . . 6  |-  ( B  =  <. x ,  y
>.  ->  ( B  e.  A  <->  <. x ,  y
>.  e.  A ) )
7 vex 2577 . . . . . . 7  |-  x  e. 
_V
8 vex 2577 . . . . . . 7  |-  y  e. 
_V
97, 8opeldm 4566 . . . . . 6  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
106, 9syl6bi 156 . . . . 5  |-  ( B  =  <. x ,  y
>.  ->  ( B  e.  A  ->  x  e.  dom  A ) )
11 inteq 3646 . . . . . . . 8  |-  ( B  =  <. x ,  y
>.  ->  |^| B  =  |^| <.
x ,  y >.
)
1211inteqd 3648 . . . . . . 7  |-  ( B  =  <. x ,  y
>.  ->  |^| |^| B  =  |^| |^|
<. x ,  y >.
)
137, 8op1stb 4237 . . . . . . 7  |-  |^| |^| <. x ,  y >.  =  x
1412, 13syl6eq 2104 . . . . . 6  |-  ( B  =  <. x ,  y
>.  ->  |^| |^| B  =  x )
1514eleq1d 2122 . . . . 5  |-  ( B  =  <. x ,  y
>.  ->  ( |^| |^| B  e.  dom  A  <->  x  e.  dom  A ) )
1610, 15sylibrd 162 . . . 4  |-  ( B  =  <. x ,  y
>.  ->  ( B  e.  A  ->  |^| |^| B  e.  dom  A ) )
1716exlimivv 1792 . . 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 119 1  |-  ( ( Rel  A  /\  B  e.  A )  ->  |^| |^| B  e.  dom  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574    C_ wss 2945   <.cop 3406   |^|cint 3643    X. cxp 4371   dom cdm 4373   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-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-int 3644  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-dm 4383
This theorem is referenced by:  1stdm  5836  fundmen  6317
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