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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 5685 | . . 3 ⊢ (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = ran (( I ↾ 𝐴) ↾ 𝐵)) |
3 | resabs1 6009 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) ↾ 𝐵) = ( I ↾ 𝐵)) | |
4 | 3 | rneqd 5934 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ran (( I ↾ 𝐴) ↾ 𝐵) = ran ( I ↾ 𝐵)) |
5 | rnresi 6072 | . . 3 ⊢ ran ( I ↾ 𝐵) = 𝐵 | |
6 | 5 | a1i 11 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ran ( I ↾ 𝐵) = 𝐵) |
7 | 2, 4, 6 | 3eqtrd 2772 | 1 ⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ⊆ wss 3945 I cid 5569 ran crn 5673 ↾ cres 5674 “ cima 5675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 |
This theorem is referenced by: fipreima 9376 psgnunilem1 19441 islinds2 21740 lindsind2 21746 ssidcn 23152 idqtop 23603 fmid 23857 ellspds 33074 rrhre 33616 sitmcl 33965 bj-imdirid 36659 bj-iminvid 36668 poimirlem15 37102 grimidvtxedg 47168 ushggricedg 47187 |
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