Theorem unielrel 6228

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 5744	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		simpr 484	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅)
3		vex 3441	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3441	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	uniopel 5461	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅)
6	5	a1i 11	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅))
7		eleq1 2821	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
8		unieq 4871	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 = ∪ ⟨𝑥, 𝑦⟩)
9	8	eleq1d 2818	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (∪ 𝐴 ∈ ∪ 𝑅 ↔ ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅))
10	6, 7, 9	3imtr4d 294	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 → ∪ 𝐴 ∈ ∪ 𝑅))
11	10	exlimivv 1933	. 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 1541  ∃wex 1780   ∈ wcel 2113  ⟨cop 4583  ∪ cuni 4860  Rel wrel 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-opab 5158  df-xp 5627  df-rel 5628

This theorem is referenced by: (None)