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Mirrors > Home > ILE Home > Th. List > dmres | GIF version |
Description: The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
dmres | ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2715 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm2 4787 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵)) |
3 | 19.41v 1882 | . . . . 5 ⊢ (∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
4 | vex 2715 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | 4 | opelres 4874 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
6 | 5 | exbii 1585 | . . . . 5 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵) ↔ ∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
7 | 1 | eldm2 4787 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
8 | 7 | anbi1i 454 | . . . . 5 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
9 | 3, 6, 8 | 3bitr4i 211 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ 𝐵)) |
10 | 2, 9 | bitr2i 184 | . . 3 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ dom (𝐴 ↾ 𝐵)) |
11 | 10 | ineqri 3301 | . 2 ⊢ (dom 𝐴 ∩ 𝐵) = dom (𝐴 ↾ 𝐵) |
12 | incom 3300 | . 2 ⊢ (dom 𝐴 ∩ 𝐵) = (𝐵 ∩ dom 𝐴) | |
13 | 11, 12 | eqtr3i 2180 | 1 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1335 ∃wex 1472 ∈ wcel 2128 ∩ cin 3101 〈cop 3564 dom cdm 4589 ↾ cres 4591 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4029 df-xp 4595 df-dm 4599 df-res 4601 |
This theorem is referenced by: ssdmres 4891 dmresexg 4892 imadisj 4951 ndmima 4966 imainrect 5034 dmresv 5047 resdmres 5080 funimacnv 5249 fnresdisj 5283 fnres 5289 ssimaex 5532 fnreseql 5580 respreima 5598 ffvresb 5633 fsnunfv 5671 funfvima 5701 offres 6086 smores 6242 smores3 6243 smores2 6244 fnfi 6884 sbthlemi5 6908 sbthlem7 6910 dmaddpi 7248 dmmulpi 7249 fvsetsid 12320 setsfun 12321 setsfun0 12322 setsresg 12324 lmres 12744 metreslem 12876 |
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