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Theorem unielrel 4845
 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 4442 . 2 ((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 simpr 103 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴𝑅)
3 vex 2560 . . . . . 6 𝑥 ∈ V
4 vex 2560 . . . . . 6 𝑦 ∈ V
53, 4uniopel 3993 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝑅𝑥, 𝑦⟩ ∈ 𝑅)
65a1i 9 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝑥, 𝑦⟩ ∈ 𝑅))
7 eleq1 2100 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
8 unieq 3589 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
98eleq1d 2106 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → ( 𝐴 𝑅𝑥, 𝑦⟩ ∈ 𝑅))
106, 7, 93imtr4d 192 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴𝑅 𝐴 𝑅))
1110exlimivv 1776 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴𝑅 𝐴 𝑅))
121, 2, 11sylc 56 1 ((Rel 𝑅𝐴𝑅) → 𝐴 𝑅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ⟨cop 3378  ∪ cuni 3580  Rel wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-opab 3819  df-xp 4351  df-rel 4352 This theorem is referenced by: (None)
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