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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 2741 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm2 4826 | . . . 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 2741 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | 4 | opelres 4913 | . . . . . 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 4826 | . . . . . 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 3329 | . 2 ⊢ (dom 𝐴 ∩ 𝐵) = dom (𝐴 ↾ 𝐵) |
12 | incom 3328 | . 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 3129 ⟨cop 3596 dom cdm 4627 ↾ cres 4629 |
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 4122 ax-pow 4175 ax-pr 4210 |
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 2740 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-br 4005 df-opab 4066 df-xp 4633 df-dm 4637 df-res 4639 |
This theorem is referenced by: ssdmres 4930 dmresexg 4931 imadisj 4991 ndmima 5006 imainrect 5075 dmresv 5088 resdmres 5121 funimacnv 5293 fnresdisj 5327 fnres 5333 ssimaex 5578 fnreseql 5627 respreima 5645 ffvresb 5680 fsnunfv 5718 funfvima 5749 offres 6136 smores 6293 smores3 6294 smores2 6295 fnfi 6936 sbthlemi5 6960 sbthlem7 6962 dmaddpi 7324 dmmulpi 7325 fvsetsid 12496 setsfun 12497 setsfun0 12498 setsresg 12500 lmres 13751 metreslem 13883 |
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