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| Description: A restriction to an image. (Contributed by NM, 29-Sep-2004.) | 
| Ref | Expression | 
|---|---|
| resima | ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | residm 6028 | . . 3 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
| 2 | 1 | rneqi 5948 | . 2 ⊢ ran ((𝐴 ↾ 𝐵) ↾ 𝐵) = ran (𝐴 ↾ 𝐵) | 
| 3 | df-ima 5698 | . 2 ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = ran ((𝐴 ↾ 𝐵) ↾ 𝐵) | |
| 4 | df-ima 5698 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
| 5 | 2, 3, 4 | 3eqtr4i 2775 | 1 ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ran crn 5686 ↾ cres 5687 “ cima 5688 | 
| 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-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 | 
| This theorem is referenced by: isarep2 6658 f1imacnv 6864 foimacnv 6865 dffv2 7004 fssrescdmd 7146 islindf4 21858 qtopres 23706 aks6d1c6lem4 42174 | 
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