![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > unielrel | Structured version Visualization version GIF version |
Description: The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.) |
Ref | Expression |
---|---|
unielrel | ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∪ 𝐴 ∈ ∪ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrel 5791 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩) | |
2 | simpr 484 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅) | |
3 | vex 3472 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3472 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | uniopel 5509 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅) |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑅)) |
7 | eleq1 2815 | . . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)) | |
8 | unieq 4913 | . . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 = ∪ ⟨𝑥, 𝑦⟩) | |
9 | 8 | eleq1d 2812 | . . . 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 1927 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 → ∪ 𝐴 ∈ ∪ 𝑅)) |
12 | 1, 2, 11 | sylc 65 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∪ 𝐴 ∈ ∪ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ⟨cop 4629 ∪ cuni 4902 Rel wrel 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-opab 5204 df-xp 5675 df-rel 5676 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |