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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 2733 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm2 4809 | . . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
3 | 1 | prid1 3689 | . . . . . 6 ⊢ 𝑥 ∈ {𝑥, 𝑦} |
4 | vex 2733 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
5 | 1, 4 | uniop 4240 | . . . . . . . . 9 ⊢ ∪ 〈𝑥, 𝑦〉 = {𝑥, 𝑦} |
6 | 1, 4 | uniopel 4241 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴) |
7 | 5, 6 | eqeltrrid 2258 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → {𝑥, 𝑦} ∈ ∪ 𝐴) |
8 | elssuni 3824 | . . . . . . . 8 ⊢ ({𝑥, 𝑦} ∈ ∪ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴) | |
9 | 7, 8 | syl 14 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴) |
10 | 9 | sseld 3146 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ ∪ ∪ 𝐴)) |
11 | 3, 10 | mpi 15 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) |
12 | 11 | exlimiv 1591 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) |
13 | 2, 12 | sylbi 120 | . . 3 ⊢ (𝑥 ∈ dom 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) |
14 | 13 | ssriv 3151 | . 2 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴 |
15 | 4 | elrn2 4853 | . . . 4 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
16 | 4 | prid2 3690 | . . . . . 6 ⊢ 𝑦 ∈ {𝑥, 𝑦} |
17 | 9 | sseld 3146 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 ∈ ∪ ∪ 𝐴)) |
18 | 16, 17 | mpi 15 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) |
19 | 18 | exlimiv 1591 | . . . 4 ⊢ (∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) |
20 | 15, 19 | sylbi 120 | . . 3 ⊢ (𝑦 ∈ ran 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) |
21 | 20 | ssriv 3151 | . 2 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 |
22 | 14, 21 | unssi 3302 | 1 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∃wex 1485 ∈ wcel 2141 ∪ cun 3119 ⊆ wss 3121 {cpr 3584 〈cop 3586 ∪ cuni 3796 dom cdm 4611 ran crn 4612 |
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 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-cnv 4619 df-dm 4621 df-rn 4622 |
This theorem is referenced by: dmexg 4875 rnexg 4876 relfld 5139 relcoi2 5141 |
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