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Theorem 1stdm 6396
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 4887 . . . . 5  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 188 . . . 4  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32sselda 3350 . . 3  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  ( _V  X.  _V ) )
4 1stval2 6366 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )
53, 4syl 16 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 1st `  A )  = 
|^| |^| A )
6 elreldm 5096 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  |^| |^| A  e.  dom  R )
75, 6eqeltrd 2512 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 1st `  A )  e. 
dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   |^|cint 4052    X. cxp 4878   dom cdm 4880   Rel wrel 4885   ` cfv 5456   1stc1st 6349
This theorem is referenced by:  frxp  6458  dprd2dlem2  15600  dprd2da  15602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fv 5464  df-1st 6351
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