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Mirrors > Home > MPE Home > Th. List > dmrnssfld | Structured version Visualization version 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 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm2 5769 | . . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
3 | 1 | prid1 4697 | . . . . . 6 ⊢ 𝑥 ∈ {𝑥, 𝑦} |
4 | vex 3497 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
5 | 1, 4 | uniop 5404 | . . . . . . . . 9 ⊢ ∪ 〈𝑥, 𝑦〉 = {𝑥, 𝑦} |
6 | 1, 4 | uniopel 5405 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴) |
7 | 5, 6 | eqeltrrid 2918 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → {𝑥, 𝑦} ∈ ∪ 𝐴) |
8 | elssuni 4867 | . . . . . . . 8 ⊢ ({𝑥, 𝑦} ∈ ∪ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴) |
10 | 9 | sseld 3965 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ ∪ ∪ 𝐴)) |
11 | 3, 10 | mpi 20 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) |
12 | 11 | exlimiv 1927 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) |
13 | 2, 12 | sylbi 219 | . . 3 ⊢ (𝑥 ∈ dom 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) |
14 | 13 | ssriv 3970 | . 2 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴 |
15 | 4 | elrn2 5820 | . . . 4 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
16 | 4 | prid2 4698 | . . . . . 6 ⊢ 𝑦 ∈ {𝑥, 𝑦} |
17 | 9 | sseld 3965 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 ∈ ∪ ∪ 𝐴)) |
18 | 16, 17 | mpi 20 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) |
19 | 18 | exlimiv 1927 | . . . 4 ⊢ (∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) |
20 | 15, 19 | sylbi 219 | . . 3 ⊢ (𝑦 ∈ ran 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) |
21 | 20 | ssriv 3970 | . 2 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 |
22 | 14, 21 | unssi 4160 | 1 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1776 ∈ wcel 2110 ∪ cun 3933 ⊆ wss 3935 {cpr 4568 〈cop 4572 ∪ cuni 4837 dom cdm 5554 ran crn 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-cnv 5562 df-dm 5564 df-rn 5565 |
This theorem is referenced by: relfld 6125 relcoi2 6127 dmexg 7612 rnexg 7613 wundm 10149 wunrn 10150 relexpdm 14401 relexprn 14405 relexpfld 14407 psdmrn 17816 dirdm 17843 dirge 17846 tailf 33723 filnetlem3 33728 |
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