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Theorem unielrel 6267
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 5791 . 2 ((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 simpr 484 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴𝑅)
3 vex 3472 . . . . . 6 𝑥 ∈ V
4 vex 3472 . . . . . 6 𝑦 ∈ V
53, 4uniopel 5509 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝑅𝑥, 𝑦⟩ ∈ 𝑅)
65a1i 11 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝑥, 𝑦⟩ ∈ 𝑅))
7 eleq1 2815 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
8 unieq 4913 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
98eleq1d 2812 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → ( 𝐴 𝑅𝑥, 𝑦⟩ ∈ 𝑅))
106, 7, 93imtr4d 294 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴𝑅 𝐴 𝑅))
1110exlimivv 1927 . 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 1533  wex 1773  wcel 2098  cop 4629   cuni 4902  Rel wrel 5674
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-opab 5204  df-xp 5675  df-rel 5676
This theorem is referenced by: (None)
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