Theorem unielrel 6166

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.

Step	Hyp	Ref	Expression
1		elrel 5697	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		simpr 484	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅)
3		vex 3426	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3426	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	uniopel 5424	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅)
6	5	a1i 11	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅))
7		eleq1 2826	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
8		unieq 4847	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 = ∪ ⟨𝑥, 𝑦⟩)
9	8	eleq1d 2823	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (∪ 𝐴 ∈ ∪ 𝑅 ↔ ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅))
10	6, 7, 9	3imtr4d 293	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 → ∪ 𝐴 ∈ ∪ 𝑅))
11	10	exlimivv 1936	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 → ∪ 𝐴 ∈ ∪ 𝑅))
12	1, 2, 11	sylc 65	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∪ 𝐴 ∈ ∪ 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ⟨cop 4564  ∪ cuni 4836  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-opab 5133  df-xp 5586  df-rel 5587

This theorem is referenced by: (None)