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Theorem unielrel 6296
Description: The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
unielrel ((Rel 𝑅𝐴𝑅) → 𝐴 𝑅)

Proof of Theorem unielrel
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrel 5811 . 2 ((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 simpr 484 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴𝑅)
3 vex 3482 . . . . . 6 𝑥 ∈ V
4 vex 3482 . . . . . 6 𝑦 ∈ V
53, 4uniopel 5526 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝑅𝑥, 𝑦⟩ ∈ 𝑅)
65a1i 11 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝑥, 𝑦⟩ ∈ 𝑅))
7 eleq1 2827 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
8 unieq 4923 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
98eleq1d 2824 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → ( 𝐴 𝑅𝑥, 𝑦⟩ ∈ 𝑅))
106, 7, 93imtr4d 294 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴𝑅 𝐴 𝑅))
1110exlimivv 1930 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴𝑅 𝐴 𝑅))
121, 2, 11sylc 65 1 ((Rel 𝑅𝐴𝑅) → 𝐴 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  cop 4637   cuni 4912  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by: (None)
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