Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  1stdm Structured version   Visualization version   GIF version

Theorem 1stdm 7106
 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 5045 . . . . 5 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 205 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
32sselda 3568 . . 3 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ (V × V))
4 1stval2 7076 . . 3 (𝐴 ∈ (V × V) → (1st𝐴) = 𝐴)
53, 4syl 17 . 2 ((Rel 𝑅𝐴𝑅) → (1st𝐴) = 𝐴)
6 elreldm 5271 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ dom 𝑅)
75, 6eqeltrd 2688 1 ((Rel 𝑅𝐴𝑅) → (1st𝐴) ∈ dom 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∩ cint 4410   × cxp 5036  dom cdm 5038  Rel wrel 5043  ‘cfv 5804  1st c1st 7057 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059 This theorem is referenced by:  frxp  7174  dprd2dlem2  18262  dprd2da  18264  gsummpt2d  29112
 Copyright terms: Public domain W3C validator