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Theorem 1stdm 5934
Description: The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
1stdm  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 1st `  A )  e. 
dom  R )

Proof of Theorem 1stdm
StepHypRef Expression
1 df-rel 4435 . . . . 5  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 118 . . . 4  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32sselda 3023 . . 3  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  ( _V  X.  _V ) )
4 1stval2 5908 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )
53, 4syl 14 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 1st `  A )  = 
|^| |^| A )
6 elreldm 4649 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  |^| |^| A  e.  dom  R )
75, 6eqeltrd 2164 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 1st `  A )  e. 
dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619    C_ wss 2997   |^|cint 3683    X. cxp 4426   dom cdm 4428   Rel wrel 4433   ` cfv 5002   1stc1st 5891
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-13 1449  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  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fv 5010  df-1st 5893
This theorem is referenced by: (None)
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