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Theorem 1stdm 8064
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 5696 . . . . 5 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 216 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
32sselda 3995 . . 3 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ (V × V))
4 1stval2 8030 . . 3 (𝐴 ∈ (V × V) → (1st𝐴) = 𝐴)
53, 4syl 17 . 2 ((Rel 𝑅𝐴𝑅) → (1st𝐴) = 𝐴)
6 elreldm 5949 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ dom 𝑅)
75, 6eqeltrd 2839 1 ((Rel 𝑅𝐴𝑅) → (1st𝐴) ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963   cint 4951   × cxp 5687  dom cdm 5689  Rel wrel 5694  cfv 6563  1st c1st 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013
This theorem is referenced by:  releldmdifi  8069  funeldmdif  8072  frxp  8150  dprd2dlem2  20075  dprd2da  20077  gsummpt2d  33035  gsumhashmul  33047  gsumwrd2dccat  33053  satfdmlem  35353  satffunlem1lem2  35388  satffunlem2lem2  35391
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