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Theorem 1stdm 7973
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 5641 . . . . 5 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 215 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
32sselda 3945 . . 3 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ (V × V))
4 1stval2 7939 . . 3 (𝐴 ∈ (V × V) → (1st𝐴) = 𝐴)
53, 4syl 17 . 2 ((Rel 𝑅𝐴𝑅) → (1st𝐴) = 𝐴)
6 elreldm 5891 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ dom 𝑅)
75, 6eqeltrd 2838 1 ((Rel 𝑅𝐴𝑅) → (1st𝐴) ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3446  wss 3911   cint 4908   × cxp 5632  dom cdm 5634  Rel wrel 5639  cfv 6497  1st c1st 7920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922
This theorem is referenced by:  releldmdifi  7978  funeldmdif  7981  frxp  8059  dprd2dlem2  19820  dprd2da  19822  gsummpt2d  31894  gsumhashmul  31901  satfdmlem  33965  satffunlem1lem2  34000  satffunlem2lem2  34003
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