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Theorem elreldm 5237
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 5015 . . . . 5  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
2 ssel 3493 . . . . 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 5067 . . . 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 2529 . . . . . 6  |-  ( B  =  <. x ,  y
>.  ->  ( B  e.  A  <->  <. x ,  y
>.  e.  A ) )
7 vex 3112 . . . . . . 7  |-  x  e. 
_V
8 vex 3112 . . . . . . 7  |-  y  e. 
_V
97, 8opeldm 5216 . . . . . 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 4726 . . . . . . 7  |-  |^| |^| <. x ,  y >.  =  x
1412, 13syl6eq 2514 . . . . . 6  |-  ( B  =  <. x ,  y
>.  ->  |^| |^| B  =  x )
1514eleq1d 2526 . . . . 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 1724 . . 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 1395   E.wex 1613    e. wcel 1819   _Vcvv 3109    C_ wss 3471   <.cop 4038   |^|cint 4288    X. cxp 5006   dom cdm 5008   Rel wrel 5013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-int 4289  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-dm 5018
This theorem is referenced by:  1stdm  6846  fundmen  7608
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