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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 2740 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm2 4825 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵)) |
3 | 19.41v 1902 | . . . . 5 ⊢ (∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
4 | vex 2740 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | 4 | opelres 4912 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
6 | 5 | exbii 1605 | . . . . 5 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
7 | 1 | eldm2 4825 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴) |
8 | 7 | anbi1i 458 | . . . . 5 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
9 | 3, 6, 8 | 3bitr4i 212 | . . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ 𝐵)) |
10 | 2, 9 | bitr2i 185 | . . 3 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ dom (𝐴 ↾ 𝐵)) |
11 | 10 | ineqri 3328 | . 2 ⊢ (dom 𝐴 ∩ 𝐵) = dom (𝐴 ↾ 𝐵) |
12 | incom 3327 | . 2 ⊢ (dom 𝐴 ∩ 𝐵) = (𝐵 ∩ dom 𝐴) | |
13 | 11, 12 | eqtr3i 2200 | 1 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∩ cin 3128 ⟨cop 3595 dom cdm 4626 ↾ cres 4628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-xp 4632 df-dm 4636 df-res 4638 |
This theorem is referenced by: ssdmres 4929 dmresexg 4930 imadisj 4990 ndmima 5005 imainrect 5074 dmresv 5087 resdmres 5120 funimacnv 5292 fnresdisj 5326 fnres 5332 ssimaex 5577 fnreseql 5626 respreima 5644 ffvresb 5679 fsnunfv 5717 funfvima 5748 offres 6135 smores 6292 smores3 6293 smores2 6294 fnfi 6935 sbthlemi5 6959 sbthlem7 6961 dmaddpi 7323 dmmulpi 7324 fvsetsid 12495 setsfun 12496 setsfun0 12497 setsresg 12499 lmres 13718 metreslem 13850 |
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