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Mirrors > Home > ILE Home > Th. List > dmrnssfld | GIF version |
Description: The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.) |
Ref | Expression |
---|---|
dmrnssfld | ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2615 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm2 4590 | . . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
3 | 1 | prid1 3522 | . . . . . 6 ⊢ 𝑥 ∈ {𝑥, 𝑦} |
4 | vex 2615 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
5 | 1, 4 | uniop 4045 | . . . . . . . . 9 ⊢ ∪ 〈𝑥, 𝑦〉 = {𝑥, 𝑦} |
6 | 1, 4 | uniopel 4046 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴) |
7 | 5, 6 | syl5eqelr 2170 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → {𝑥, 𝑦} ∈ ∪ 𝐴) |
8 | elssuni 3655 | . . . . . . . 8 ⊢ ({𝑥, 𝑦} ∈ ∪ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴) | |
9 | 7, 8 | syl 14 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴) |
10 | 9 | sseld 3009 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ ∪ ∪ 𝐴)) |
11 | 3, 10 | mpi 15 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) |
12 | 11 | exlimiv 1530 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) |
13 | 2, 12 | sylbi 119 | . . 3 ⊢ (𝑥 ∈ dom 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) |
14 | 13 | ssriv 3014 | . 2 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴 |
15 | 4 | elrn2 4633 | . . . 4 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
16 | 4 | prid2 3523 | . . . . . 6 ⊢ 𝑦 ∈ {𝑥, 𝑦} |
17 | 9 | sseld 3009 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 ∈ ∪ ∪ 𝐴)) |
18 | 16, 17 | mpi 15 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) |
19 | 18 | exlimiv 1530 | . . . 4 ⊢ (∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) |
20 | 15, 19 | sylbi 119 | . . 3 ⊢ (𝑦 ∈ ran 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) |
21 | 20 | ssriv 3014 | . 2 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 |
22 | 14, 21 | unssi 3159 | 1 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∃wex 1422 ∈ wcel 1434 ∪ cun 2982 ⊆ wss 2984 {cpr 3423 〈cop 3425 ∪ cuni 3627 dom cdm 4399 ran crn 4400 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-cnv 4407 df-dm 4409 df-rn 4410 |
This theorem is referenced by: dmexg 4653 rnexg 4654 relfld 4911 relcoi2 4913 |
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