![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > resiima | Structured version Visualization version GIF version |
Description: The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.) |
Ref | Expression |
---|---|
resiima | ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5702 | . . 3 ⊢ (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵)) |
3 | resabs1 6027 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) ↾ 𝐵) = ( I ↾ 𝐵)) | |
4 | 3 | rneqd 5952 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ran (( I ↾ 𝐴) ↾ 𝐵) = ran ( I ↾ 𝐵)) |
5 | rnresi 6095 | . . 3 ⊢ ran ( I ↾ 𝐵) = 𝐵 | |
6 | 5 | a1i 11 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ran ( I ↾ 𝐵) = 𝐵) |
7 | 2, 4, 6 | 3eqtrd 2779 | 1 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ⊆ wss 3963 I cid 5582 ran crn 5690 ↾ cres 5691 “ cima 5692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: fipreima 9396 psgnunilem1 19526 islinds2 21851 lindsind2 21857 ssidcn 23279 idqtop 23730 fmid 23984 ellspds 33376 rrhre 33984 sitmcl 34333 bj-imdirid 37169 bj-iminvid 37178 poimirlem15 37622 grimidvtxedg 47814 ushggricedg 47834 |
Copyright terms: Public domain | W3C validator |