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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 4008 | . . 3 ⊢ 𝐴 ⊆ V | |
| 2 | dmi 5932 | . . 3 ⊢ dom I = V | |
| 3 | 1, 2 | sseqtrri 4033 | . 2 ⊢ 𝐴 ⊆ dom I |
| 4 | ssdmres 6031 | . 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 1540 Vcvv 3480 ⊆ wss 3951 I cid 5577 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-id 5578 df-xp 5691 df-rel 5692 df-dm 5695 df-res 5697 |
| This theorem is referenced by: iordsmo 8397 residfi 9378 hartogslem1 9582 dfac9 10177 hsmexlem5 10470 relexpdmg 15081 relexpfld 15088 relexpaddg 15092 dirdm 18645 islinds2 21833 lindsind2 21839 f1linds 21845 wilthlem3 27113 ausgrusgrb 29182 usgrres1 29332 usgrexilem 29457 filnetlem3 36381 filnetlem4 36382 rclexi 43628 dfrtrcl5 43642 dfrcl2 43687 brfvrcld2 43705 iunrelexp0 43715 relexpiidm 43717 relexp01min 43726 ushggricedg 47896 stgrusgra 47926 gpgusgra 48012 uspgrsprfo 48064 |
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