![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3444 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm2 5734 | . . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
3 | 1 | prid1 4658 | . . . . . 6 ⊢ 𝑥 ∈ {𝑥, 𝑦} |
4 | vex 3444 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
5 | 1, 4 | uniop 5370 | . . . . . . . . 9 ⊢ ∪ 〈𝑥, 𝑦〉 = {𝑥, 𝑦} |
6 | 1, 4 | uniopel 5371 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∪ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴) |
7 | 5, 6 | eqeltrrid 2895 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → {𝑥, 𝑦} ∈ ∪ 𝐴) |
8 | elssuni 4830 | . . . . . . . 8 ⊢ ({𝑥, 𝑦} ∈ ∪ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴) |
10 | 9 | sseld 3914 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ ∪ ∪ 𝐴)) |
11 | 3, 10 | mpi 20 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) |
12 | 11 | exlimiv 1931 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) |
13 | 2, 12 | sylbi 220 | . . 3 ⊢ (𝑥 ∈ dom 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴) |
14 | 13 | ssriv 3919 | . 2 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴 |
15 | 4 | elrn2 5785 | . . . 4 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
16 | 4 | prid2 4659 | . . . . . 6 ⊢ 𝑦 ∈ {𝑥, 𝑦} |
17 | 9 | sseld 3914 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 ∈ ∪ ∪ 𝐴)) |
18 | 16, 17 | mpi 20 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) |
19 | 18 | exlimiv 1931 | . . . 4 ⊢ (∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) |
20 | 15, 19 | sylbi 220 | . . 3 ⊢ (𝑦 ∈ ran 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴) |
21 | 20 | ssriv 3919 | . 2 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 |
22 | 14, 21 | unssi 4112 | 1 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1781 ∈ wcel 2111 ∪ cun 3879 ⊆ wss 3881 {cpr 4527 〈cop 4531 ∪ cuni 4800 dom cdm 5519 ran crn 5520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: relfld 6094 relcoi2 6096 dmexg 7594 rnexg 7595 wundm 10139 wunrn 10140 relexpdm 14394 relexprn 14398 relexpfld 14400 psdmrn 17809 dirdm 17836 dirge 17839 tailf 33836 filnetlem3 33841 |
Copyright terms: Public domain | W3C validator |