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Theorem 1stdm 6083
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 𝑅𝐴𝑅) → (1st𝐴) ∈ dom 𝑅)

Proof of Theorem 1stdm
StepHypRef Expression
1 df-rel 4549 . . . . 5 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 119 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
32sselda 3097 . . 3 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ (V × V))
4 1stval2 6056 . . 3 (𝐴 ∈ (V × V) → (1st𝐴) = 𝐴)
53, 4syl 14 . 2 ((Rel 𝑅𝐴𝑅) → (1st𝐴) = 𝐴)
6 elreldm 4768 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ dom 𝑅)
75, 6eqeltrd 2216 1 ((Rel 𝑅𝐴𝑅) → (1st𝐴) ∈ dom 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  Vcvv 2686  wss 3071   cint 3774   × cxp 4540  dom cdm 4542  Rel wrel 4547  cfv 5126  1st c1st 6039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-br 3933  df-opab 3993  df-mpt 3994  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-iota 5091  df-fun 5128  df-fv 5134  df-1st 6041
This theorem is referenced by: (None)
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