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Theorem 1stdm 5836
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 4380 . . . . 5  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 117 . . . 4  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32sselda 2973 . . 3  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  ( _V  X.  _V ) )
4 1stval2 5810 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )
53, 4syl 14 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 1st `  A )  = 
|^| |^| A )
6 elreldm 4588 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  |^| |^| A  e.  dom  R )
75, 6eqeltrd 2130 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  ( 1st `  A )  e. 
dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   _Vcvv 2574    C_ wss 2945   |^|cint 3643    X. cxp 4371   dom cdm 4373   Rel wrel 4378   ` cfv 4930   1stc1st 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fv 4938  df-1st 5795
This theorem is referenced by: (None)
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