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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 2684 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm2 4732 | . . . 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 2684 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | 4 | opelres 4819 | . . . . . 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 4732 | . . . . . 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 3264 | . 2 ⊢ (dom 𝐴 ∩ 𝐵) = dom (𝐴 ↾ 𝐵) |
12 | incom 3263 | . 2 ⊢ (dom 𝐴 ∩ 𝐵) = (𝐵 ∩ dom 𝐴) | |
13 | 11, 12 | eqtr3i 2160 | 1 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∩ cin 3065 〈cop 3525 dom cdm 4534 ↾ cres 4536 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-dm 4544 df-res 4546 |
This theorem is referenced by: ssdmres 4836 dmresexg 4837 imadisj 4896 ndmima 4911 imainrect 4979 dmresv 4992 resdmres 5025 funimacnv 5194 fnresdisj 5228 fnres 5234 ssimaex 5475 fnreseql 5523 respreima 5541 ffvresb 5576 fsnunfv 5614 funfvima 5642 offres 6026 smores 6182 smores3 6183 smores2 6184 fnfi 6818 sbthlemi5 6842 sbthlem7 6844 dmaddpi 7126 dmmulpi 7127 fvsetsid 11982 setsfun 11983 setsfun0 11984 setsresg 11986 lmres 12406 metreslem 12538 |
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