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Mirrors > Home > MPE Home > Th. List > dmresi | Structured version Visualization version GIF version |
Description: The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.) |
Ref | Expression |
---|---|
dmresi | ⊢ dom ( I ↾ 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 4019 | . . 3 ⊢ 𝐴 ⊆ V | |
2 | dmi 5934 | . . 3 ⊢ dom I = V | |
3 | 1, 2 | sseqtrri 4032 | . 2 ⊢ 𝐴 ⊆ dom I |
4 | ssdmres 6032 | . 2 ⊢ (𝐴 ⊆ dom I ↔ dom ( I ↾ 𝐴) = 𝐴) | |
5 | 3, 4 | mpbi 230 | 1 ⊢ dom ( I ↾ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 Vcvv 3477 ⊆ wss 3962 I cid 5581 dom cdm 5688 ↾ cres 5690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-dm 5698 df-res 5700 |
This theorem is referenced by: iordsmo 8395 residfi 9375 hartogslem1 9579 dfac9 10174 hsmexlem5 10467 relexpdmg 15077 relexpfld 15084 relexpaddg 15088 dirdm 18657 islinds2 21850 lindsind2 21856 f1linds 21862 wilthlem3 27127 ausgrusgrb 29196 usgrres1 29346 usgrexilem 29471 filnetlem3 36362 filnetlem4 36363 rclexi 43604 dfrtrcl5 43618 dfrcl2 43663 brfvrcld2 43681 iunrelexp0 43691 relexpiidm 43693 relexp01min 43702 ushggricedg 47833 stgrusgra 47861 gpgusgra 47946 uspgrsprfo 47991 |
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