Theorem unielrel 5577

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 5145	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		simpr 476	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅)
3		vex 3176	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3176	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	uniopel 4903	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅)
6	5	a1i 11	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅))
7		eleq1 2676	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
8		unieq 4380	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 = ∪ ⟨𝑥, 𝑦⟩)
9	8	eleq1d 2672	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (∪ 𝐴 ∈ ∪ 𝑅 ↔ ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅))
10	6, 7, 9	3imtr4d 282	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 → ∪ 𝐴 ∈ ∪ 𝑅))
11	10	exlimivv 1847	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 → ∪ 𝐴 ∈ ∪ 𝑅))
12	1, 2, 11	sylc 63	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∪ 𝐴 ∈ ∪ 𝑅)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ⟨cop 4131  ∪ cuni 4372  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-opab 4644  df-xp 5044  df-rel 5045

This theorem is referenced by: (None)