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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 2689 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm2 4737 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵)) |
3 | 19.41v 1874 | . . . . 5 ⊢ (∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
4 | vex 2689 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | 4 | opelres 4824 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
6 | 5 | exbii 1584 | . . . . 5 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵) ↔ ∃𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
7 | 1 | eldm2 4737 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
8 | 7 | anbi1i 453 | . . . . 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 3269 | . 2 ⊢ (dom 𝐴 ∩ 𝐵) = dom (𝐴 ↾ 𝐵) |
12 | incom 3268 | . 2 ⊢ (dom 𝐴 ∩ 𝐵) = (𝐵 ∩ dom 𝐴) | |
13 | 11, 12 | eqtr3i 2162 | 1 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∩ cin 3070 〈cop 3530 dom cdm 4539 ↾ cres 4541 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-dm 4549 df-res 4551 |
This theorem is referenced by: ssdmres 4841 dmresexg 4842 imadisj 4901 ndmima 4916 imainrect 4984 dmresv 4997 resdmres 5030 funimacnv 5199 fnresdisj 5233 fnres 5239 ssimaex 5482 fnreseql 5530 respreima 5548 ffvresb 5583 fsnunfv 5621 funfvima 5649 offres 6033 smores 6189 smores3 6190 smores2 6191 fnfi 6825 sbthlemi5 6849 sbthlem7 6851 dmaddpi 7133 dmmulpi 7134 fvsetsid 11993 setsfun 11994 setsfun0 11995 setsresg 11997 lmres 12417 metreslem 12549 |
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