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Theorem unielrel 4788
 Description: The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
unielrel ((Rel 𝑅 A 𝑅) → A 𝑅)

Proof of Theorem unielrel
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrel 4385 . 2 ((Rel 𝑅 A 𝑅) → xy A = ⟨x, y⟩)
2 simpr 103 . 2 ((Rel 𝑅 A 𝑅) → A 𝑅)
3 vex 2554 . . . . . 6 x V
4 vex 2554 . . . . . 6 y V
53, 4uniopel 3984 . . . . 5 (⟨x, y 𝑅x, y 𝑅)
65a1i 9 . . . 4 (A = ⟨x, y⟩ → (⟨x, y 𝑅x, y 𝑅))
7 eleq1 2097 . . . 4 (A = ⟨x, y⟩ → (A 𝑅 ↔ ⟨x, y 𝑅))
8 unieq 3580 . . . . 5 (A = ⟨x, y⟩ → A = x, y⟩)
98eleq1d 2103 . . . 4 (A = ⟨x, y⟩ → ( A 𝑅x, y 𝑅))
106, 7, 93imtr4d 192 . . 3 (A = ⟨x, y⟩ → (A 𝑅 A 𝑅))
1110exlimivv 1773 . 2 (xy A = ⟨x, y⟩ → (A 𝑅 A 𝑅))
121, 2, 11sylc 56 1 ((Rel 𝑅 A 𝑅) → A 𝑅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ⟨cop 3370  ∪ cuni 3571  Rel wrel 4293 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-opab 3810  df-xp 4294  df-rel 4295 This theorem is referenced by: (None)
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