Theorem unielrel 4992

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 4569	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		simpr 109	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅)
3		vex 2636	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 2636	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	uniopel 4107	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅)
6	5	a1i 9	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅))
7		eleq1 2157	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
8		unieq 3684	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 = ∪ ⟨𝑥, 𝑦⟩)
9	8	eleq1d 2163	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (∪ 𝐴 ∈ ∪ 𝑅 ↔ ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅))
10	6, 7, 9	3imtr4d 202	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 → ∪ 𝐴 ∈ ∪ 𝑅))
11	10	exlimivv 1831	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 → ∪ 𝐴 ∈ ∪ 𝑅))
12	1, 2, 11	sylc 62	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∪ 𝐴 ∈ ∪ 𝑅)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1296  ∃wex 1433   ∈ wcel 1445  ⟨cop 3469  ∪ cuni 3675  Rel wrel 4472

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-opab 3922  df-xp 4473  df-rel 4474

This theorem is referenced by: (None)