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Theorem elreldm 5941
Description: The first member of an ordered pair in a relation belongs to the domain of the relation (see op1stb 5477). (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 5689 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
2 ssel 3975 . . . . 5 (𝐴 ⊆ (V × V) → (𝐵𝐴𝐵 ∈ (V × V)))
31, 2sylbi 216 . . . 4 (Rel 𝐴 → (𝐵𝐴𝐵 ∈ (V × V)))
4 elvv 5756 . . . 4 (𝐵 ∈ (V × V) ↔ ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
53, 4imbitrdi 250 . . 3 (Rel 𝐴 → (𝐵𝐴 → ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩))
6 eleq1 2817 . . . . . 6 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
7 vex 3477 . . . . . . 7 𝑥 ∈ V
8 vex 3477 . . . . . . 7 𝑦 ∈ V
97, 8opeldm 5914 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
106, 9biimtrdi 252 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴𝑥 ∈ dom 𝐴))
11 inteq 4956 . . . . . . . 8 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥, 𝑦⟩)
1211inteqd 4958 . . . . . . 7 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥, 𝑦⟩)
137, 8op1stb 5477 . . . . . . 7 𝑥, 𝑦⟩ = 𝑥
1412, 13eqtrdi 2784 . . . . . 6 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥)
1514eleq1d 2814 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → ( 𝐵 ∈ dom 𝐴𝑥 ∈ dom 𝐴))
1610, 15sylibrd 258 . . . 4 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 𝐵 ∈ dom 𝐴))
1716exlimivv 1927 . . 3 (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 𝐵 ∈ dom 𝐴))
185, 17syli 39 . 2 (Rel 𝐴 → (𝐵𝐴 𝐵 ∈ dom 𝐴))
1918imp 405 1 ((Rel 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3473  wss 3949  cop 4638   cint 4953   × cxp 5680  dom cdm 5682  Rel wrel 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-int 4954  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-dm 5692
This theorem is referenced by:  1stdm  8050  fundmen  9062
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