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| 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 5807 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
| 2 | simpr 484 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅) | |
| 3 | vex 3483 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 4 | vex 3483 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 5 | 3, 4 | uniopel 5520 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝑅 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅) | 
| 6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (〈𝑥, 𝑦〉 ∈ 𝑅 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅)) | 
| 7 | eleq1 2828 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
| 8 | unieq 4917 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 = ∪ 〈𝑥, 𝑦〉) | |
| 9 | 8 | eleq1d 2825 | . . . 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 1931 | . 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 1539 ∃wex 1778 ∈ wcel 2107 〈cop 4631 ∪ cuni 4906 Rel wrel 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-opab 5205 df-xp 5690 df-rel 5691 | 
| This theorem is referenced by: (None) | 
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