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| Mirrors > Home > MPE Home > Th. List > dmres | Structured version Visualization version 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 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 2 | 1 | eldm2 5912 | . . . 4 ⊢ (𝑥 ∈ dom (𝐴 ↾ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵)) |
| 3 | 19.42v 1953 | . . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴)) | |
| 4 | vex 3484 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 5 | 4 | opelresi 6005 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) |
| 6 | 5 | exbii 1848 | . . . . 5 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) |
| 7 | 1 | eldm2 5912 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
| 8 | 7 | anbi2i 623 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴)) |
| 9 | 3, 6, 8 | 3bitr4i 303 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐴)) |
| 10 | 2, 9 | bitr2i 276 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐴) ↔ 𝑥 ∈ dom (𝐴 ↾ 𝐵)) |
| 11 | 10 | ineqri 4212 | . 2 ⊢ (𝐵 ∩ dom 𝐴) = dom (𝐴 ↾ 𝐵) |
| 12 | 11 | eqcomi 2746 | 1 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∩ cin 3950 〈cop 4632 dom cdm 5685 ↾ cres 5687 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 df-res 5697 |
| This theorem is referenced by: ssdmres 6031 dmresexg 6032 dmressnsn 6041 eldmeldmressn 6043 relresdm1 6051 imadisj 6098 imainrect 6201 dmresv 6220 resdmres 6252 resdmss 6255 coeq0 6275 resssxp 6290 snres0 6318 funimacnv 6647 fnresdisj 6688 fnres 6695 fresaunres2 6780 nfvres 6947 ssimaex 6994 fnreseql 7068 respreima 7086 fveqressseq 7099 ffvresb 7145 fsnunfv 7207 funfvima 7250 funiunfv 7268 offres 8008 fnwelem 8156 ressuppss 8208 ressuppssdif 8210 frrlem11 8321 frrlem12 8322 smores 8392 smores3 8393 smores2 8394 tz7.44-2 8447 tz7.44-3 8448 frfnom 8475 sbthlem5 9127 sbthlem7 9129 domss2 9176 imafi 9353 ordtypelem4 9561 wdomima2g 9626 r0weon 10052 imadomg 10574 dmaddpi 10930 dmmulpi 10931 ltweuz 14002 dmhashres 14380 limsupgle 15513 fvsetsid 17205 setsdm 17207 setsfun 17208 setsfun0 17209 setsres 17215 lubdm 18396 glbdm 18409 gsumzaddlem 19939 dprdcntz2 20058 lmres 23308 imacmp 23405 qtoptop2 23707 kqdisj 23740 metreslem 24372 setsmstopn 24490 ismbl 25561 mbfres 25679 dvres3a 25949 cpnres 25973 dvlipcn 26033 dvlip2 26034 c1lip3 26038 dvcnvrelem1 26056 dvcvx 26059 dvlog 26693 sltres 27707 nolesgn2ores 27717 nogesgn1ores 27719 nodense 27737 nosupres 27752 nosupbnd1lem1 27753 nosupbnd2lem1 27760 nosupbnd2 27761 noinfres 27767 noinfbnd1lem1 27768 noinfbnd2lem1 27775 noetasuplem2 27779 noetainflem2 27783 uhgrspansubgrlem 29307 trlsegvdeglem4 30242 hlimcaui 31255 ftc2re 34613 dfrdg2 35796 bj-fvsnun2 37257 caures 37767 ssbnd 37795 mapfzcons1 42728 diophrw 42770 eldioph2lem1 42771 eldioph2lem2 42772 tfsconcatrev 43361 limsupresxr 45781 liminfresxr 45782 fourierdlem93 46214 fouriersw 46246 sssmf 46753 eldmressn 47049 fnresfnco 47053 afvres 47184 afv2res 47251 resinsn 48772 resinsnALT 48773 tposrescnv 48779 |
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