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Theorem 1stdm 6183
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 4712 . . . . 5  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 186 . . . 4  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32sselda 3193 . . 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 15 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 1st `  A )  = 
|^| |^| A )
6 elreldm 4919 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  |^| |^| A  e.  dom  R )
75, 6eqeltrd 2370 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 1st `  A )  e. 
dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   |^|cint 3878    X. cxp 4703   dom cdm 4705   Rel wrel 4710   ` cfv 5271   1stc1st 6136
This theorem is referenced by:  frxp  6241  dprd2dlem2  15291  dprd2da  15293  11st22nd  25148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-1st 6138
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