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Theorem 1stdm 6180
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 4632 . . . . 5  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 120 . . . 4  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32sselda 3155 . . 3  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  ( _V  X.  _V ) )
4 1stval2 6153 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )
53, 4syl 14 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 1st `  A )  = 
|^| |^| A )
6 elreldm 4852 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  |^| |^| A  e.  dom  R )
75, 6eqeltrd 2254 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 1st `  A )  e. 
dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   _Vcvv 2737    C_ wss 3129   |^|cint 3844    X. cxp 4623   dom cdm 4625   Rel wrel 4630   ` cfv 5215   1stc1st 6136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5177  df-fun 5217  df-fv 5223  df-1st 6138
This theorem is referenced by: (None)
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