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Theorem elreldm 5233
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 5012 . . . . 5  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
2 ssel 3503 . . . . 5  |-  ( A 
C_  ( _V  X.  _V )  ->  ( B  e.  A  ->  B  e.  ( _V  X.  _V ) ) )
31, 2sylbi 195 . . . 4  |-  ( Rel 
A  ->  ( B  e.  A  ->  B  e.  ( _V  X.  _V ) ) )
4 elvv 5064 . . . 4  |-  ( B  e.  ( _V  X.  _V )  <->  E. x E. y  B  =  <. x ,  y >. )
53, 4syl6ib 226 . . 3  |-  ( Rel 
A  ->  ( B  e.  A  ->  E. x E. y  B  =  <. x ,  y >.
) )
6 eleq1 2539 . . . . . 6  |-  ( B  =  <. x ,  y
>.  ->  ( B  e.  A  <->  <. x ,  y
>.  e.  A ) )
7 vex 3121 . . . . . . 7  |-  x  e. 
_V
8 vex 3121 . . . . . . 7  |-  y  e. 
_V
97, 8opeldm 5212 . . . . . 6  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
106, 9syl6bi 228 . . . . 5  |-  ( B  =  <. x ,  y
>.  ->  ( B  e.  A  ->  x  e.  dom  A ) )
11 inteq 4291 . . . . . . . 8  |-  ( B  =  <. x ,  y
>.  ->  |^| B  =  |^| <.
x ,  y >.
)
1211inteqd 4293 . . . . . . 7  |-  ( B  =  <. x ,  y
>.  ->  |^| |^| B  =  |^| |^|
<. x ,  y >.
)
137, 8op1stb 4723 . . . . . . 7  |-  |^| |^| <. x ,  y >.  =  x
1412, 13syl6eq 2524 . . . . . 6  |-  ( B  =  <. x ,  y
>.  ->  |^| |^| B  =  x )
1514eleq1d 2536 . . . . 5  |-  ( B  =  <. x ,  y
>.  ->  ( |^| |^| B  e.  dom  A  <->  x  e.  dom  A ) )
1610, 15sylibrd 234 . . . 4  |-  ( B  =  <. x ,  y
>.  ->  ( B  e.  A  ->  |^| |^| B  e.  dom  A ) )
1716exlimivv 1699 . . 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 429 1  |-  ( ( Rel  A  /\  B  e.  A )  ->  |^| |^| B  e.  dom  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3118    C_ wss 3481   <.cop 4039   |^|cint 4288    X. cxp 5003   dom cdm 5005   Rel wrel 5010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-int 4289  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-dm 5015
This theorem is referenced by:  1stdm  6842  fundmen  7601
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