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Theorem 1stdm 8044
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 5685 . . . . 5 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 215 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
32sselda 3980 . . 3 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ (V × V))
4 1stval2 8010 . . 3 (𝐴 ∈ (V × V) → (1st𝐴) = 𝐴)
53, 4syl 17 . 2 ((Rel 𝑅𝐴𝑅) → (1st𝐴) = 𝐴)
6 elreldm 5937 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ dom 𝑅)
75, 6eqeltrd 2829 1 ((Rel 𝑅𝐴𝑅) → (1st𝐴) ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  Vcvv 3471  wss 3947   cint 4949   × cxp 5676  dom cdm 5678  Rel wrel 5683  cfv 6548  1st c1st 7991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-1st 7993
This theorem is referenced by:  releldmdifi  8049  funeldmdif  8052  frxp  8131  dprd2dlem2  19997  dprd2da  19999  gsummpt2d  32776  gsumhashmul  32783  satfdmlem  34978  satffunlem1lem2  35013  satffunlem2lem2  35016
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