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Theorem unielrel 5822
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 5380 . 2 ((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 simpr 479 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴𝑅)
3 vex 3344 . . . . . 6 𝑥 ∈ V
4 vex 3344 . . . . . 6 𝑦 ∈ V
53, 4uniopel 5127 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝑅𝑥, 𝑦⟩ ∈ 𝑅)
65a1i 11 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝑥, 𝑦⟩ ∈ 𝑅))
7 eleq1 2828 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
8 unieq 4597 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
98eleq1d 2825 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → ( 𝐴 𝑅𝑥, 𝑦⟩ ∈ 𝑅))
106, 7, 93imtr4d 283 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴𝑅 𝐴 𝑅))
1110exlimivv 2010 . 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 383   = wceq 1632  wex 1853  wcel 2140  cop 4328   cuni 4589  Rel wrel 5272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-rex 3057  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-opab 4866  df-xp 5273  df-rel 5274
This theorem is referenced by: (None)
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