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Mirrors > Home > ILE Home > Th. List > unielrel | 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 4711 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | simpr 109 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅) | |
3 | vex 2733 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 2733 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | uniopel 4239 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝑅 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅) |
6 | 5 | a1i 9 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (〈𝑥, 𝑦〉 ∈ 𝑅 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅)) |
7 | eleq1 2233 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
8 | unieq 3803 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∪ 𝐴 = ∪ 〈𝑥, 𝑦〉) | |
9 | 8 | eleq1d 2239 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (∪ 𝐴 ∈ ∪ 𝑅 ↔ ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝑅)) |
10 | 6, 7, 9 | 3imtr4d 202 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝑅 → ∪ 𝐴 ∈ ∪ 𝑅)) |
11 | 10 | exlimivv 1889 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝑅 → ∪ 𝐴 ∈ ∪ 𝑅)) |
12 | 1, 2, 11 | sylc 62 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∪ 𝐴 ∈ ∪ 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∃wex 1485 ∈ wcel 2141 〈cop 3584 ∪ cuni 3794 Rel wrel 4614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-opab 4049 df-xp 4615 df-rel 4616 |
This theorem is referenced by: (None) |
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