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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 5673 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | simpr 487 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅) | |
3 | vex 3499 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3499 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | uniopel 5408 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝑅 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅) |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (〈𝑥, 𝑦〉 ∈ 𝑅 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅)) |
7 | eleq1 2902 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
8 | unieq 4851 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 = ∪ 〈𝑥, 𝑦〉) | |
9 | 8 | eleq1d 2899 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (∪ 𝐴 ∈ ∪ 𝑅 ↔ ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅)) |
10 | 6, 7, 9 | 3imtr4d 296 | . . 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 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 〈cop 4575 ∪ cuni 4840 Rel wrel 5562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-opab 5131 df-xp 5563 df-rel 5564 |
This theorem is referenced by: (None) |
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