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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 5635 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | simpr 488 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅) | |
3 | vex 3444 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3444 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | uniopel 5371 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝑅 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅) |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (〈𝑥, 𝑦〉 ∈ 𝑅 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅)) |
7 | eleq1 2877 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
8 | unieq 4811 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 = ∪ 〈𝑥, 𝑦〉) | |
9 | 8 | eleq1d 2874 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (∪ 𝐴 ∈ ∪ 𝑅 ↔ ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅)) |
10 | 6, 7, 9 | 3imtr4d 297 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝑅 → ∪ 𝐴 ∈ ∪ 𝑅)) |
11 | 10 | exlimivv 1933 | . 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 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 〈cop 4531 ∪ cuni 4800 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-opab 5093 df-xp 5525 df-rel 5526 |
This theorem is referenced by: (None) |
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