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Theorem elreldm 5271
Description: The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
Assertion
Ref Expression
elreldm ((Rel 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐴)

Proof of Theorem elreldm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rel 5045 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
2 ssel 3562 . . . . 5 (𝐴 ⊆ (V × V) → (𝐵𝐴𝐵 ∈ (V × V)))
31, 2sylbi 206 . . . 4 (Rel 𝐴 → (𝐵𝐴𝐵 ∈ (V × V)))
4 elvv 5100 . . . 4 (𝐵 ∈ (V × V) ↔ ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
53, 4syl6ib 240 . . 3 (Rel 𝐴 → (𝐵𝐴 → ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩))
6 eleq1 2676 . . . . . 6 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
7 vex 3176 . . . . . . 7 𝑥 ∈ V
8 vex 3176 . . . . . . 7 𝑦 ∈ V
97, 8opeldm 5250 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
106, 9syl6bi 242 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴𝑥 ∈ dom 𝐴))
11 inteq 4413 . . . . . . . 8 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥, 𝑦⟩)
1211inteqd 4415 . . . . . . 7 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥, 𝑦⟩)
137, 8op1stb 4867 . . . . . . 7 𝑥, 𝑦⟩ = 𝑥
1412, 13syl6eq 2660 . . . . . 6 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥)
1514eleq1d 2672 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → ( 𝐵 ∈ dom 𝐴𝑥 ∈ dom 𝐴))
1610, 15sylibrd 248 . . . 4 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 𝐵 ∈ dom 𝐴))
1716exlimivv 1847 . . 3 (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 𝐵 ∈ dom 𝐴))
185, 17syli 38 . 2 (Rel 𝐴 → (𝐵𝐴 𝐵 ∈ dom 𝐴))
1918imp 444 1 ((Rel 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wex 1695  wcel 1977  Vcvv 3173  wss 3540  cop 4131   cint 4410   × cxp 5036  dom cdm 5038  Rel wrel 5043
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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  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-int 4411  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048
This theorem is referenced by:  1stdm  7106  fundmen  7916
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