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| 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 3483 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 2 | 1 | eldm2 5911 | . . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) | 
| 3 | 1 | prid1 4761 | . . . . . 6 ⊢ 𝑥 ∈ {𝑥, 𝑦} | 
| 4 | vex 3483 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
| 5 | 1, 4 | uniop 5519 | . . . . . . . . 9 ⊢ ∪ 〈𝑥, 𝑦〉 = {𝑥, 𝑦} | 
| 6 | 1, 4 | uniopel 5520 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴) | 
| 7 | 5, 6 | eqeltrrid 2845 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → {𝑥, 𝑦} ∈ ∪ 𝐴) | 
| 8 | elssuni 4936 | . . . . . . . 8 ⊢ ({𝑥, 𝑦} ∈ ∪ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴) | |
| 9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴) | 
| 10 | 9 | sseld 3981 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ ∪ ∪ 𝐴)) | 
| 11 | 3, 10 | mpi 20 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) | 
| 12 | 11 | exlimiv 1929 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) | 
| 13 | 2, 12 | sylbi 217 | . . 3 ⊢ (𝑥 ∈ dom 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) | 
| 14 | 13 | ssriv 3986 | . 2 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴 | 
| 15 | 4 | elrn2 5902 | . . . 4 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) | 
| 16 | 4 | prid2 4762 | . . . . . 6 ⊢ 𝑦 ∈ {𝑥, 𝑦} | 
| 17 | 9 | sseld 3981 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 ∈ ∪ ∪ 𝐴)) | 
| 18 | 16, 17 | mpi 20 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) | 
| 19 | 18 | exlimiv 1929 | . . . 4 ⊢ (∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) | 
| 20 | 15, 19 | sylbi 217 | . . 3 ⊢ (𝑦 ∈ ran 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) | 
| 21 | 20 | ssriv 3986 | . 2 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 | 
| 22 | 14, 21 | unssi 4190 | 1 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∃wex 1778 ∈ wcel 2107 ∪ cun 3948 ⊆ wss 3950 {cpr 4627 〈cop 4631 ∪ cuni 4906 dom cdm 5684 ran crn 5685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 | 
| This theorem is referenced by: relfld 6294 relcoi2 6296 dmexg 7924 rnexg 7925 wundm 10769 wunrn 10770 relexpdm 15083 relexprn 15087 relexpfld 15089 psdmrn 18619 dirdm 18646 dirge 18649 tailf 36377 filnetlem3 36382 dmwf 44987 rnwf 44988 | 
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