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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 5650 | . . 3 ⊢ (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵)) |
3 | resabs1 5971 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) ↾ 𝐵) = ( I ↾ 𝐵)) | |
4 | 3 | rneqd 5897 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ran (( I ↾ 𝐴) ↾ 𝐵) = ran ( I ↾ 𝐵)) |
5 | rnresi 6031 | . . 3 ⊢ ran ( I ↾ 𝐵) = 𝐵 | |
6 | 5 | a1i 11 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ran ( I ↾ 𝐵) = 𝐵) |
7 | 2, 4, 6 | 3eqtrd 2777 | 1 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ⊆ wss 3914 I cid 5534 ran crn 5638 ↾ cres 5639 “ cima 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 |
This theorem is referenced by: fipreima 9308 psgnunilem1 19283 islinds2 21242 lindsind2 21248 ssidcn 22629 idqtop 23080 fmid 23334 ellspds 32211 rrhre 32666 sitmcl 33015 bj-imdirid 35707 bj-iminvid 35716 poimirlem15 36143 isomgreqve 46107 ushrisomgr 46123 |
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