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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 5811 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | simpr 484 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅) | |
3 | vex 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3482 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | uniopel 5526 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝑅 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅) |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (〈𝑥, 𝑦〉 ∈ 𝑅 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅)) |
7 | eleq1 2827 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
8 | unieq 4923 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 = ∪ 〈𝑥, 𝑦〉) | |
9 | 8 | eleq1d 2824 | . . . 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 1930 | . 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 1537 ∃wex 1776 ∈ wcel 2106 〈cop 4637 ∪ cuni 4912 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: (None) |
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