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Mirrors > Home > MPE Home > Th. List > resima | Structured version Visualization version GIF version |
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 5350 | . . 3 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
2 | 1 | rneqi 5273 | . 2 ⊢ ran ((𝐴 ↾ 𝐵) ↾ 𝐵) = ran (𝐴 ↾ 𝐵) |
3 | df-ima 5051 | . 2 ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = ran ((𝐴 ↾ 𝐵) ↾ 𝐵) | |
4 | df-ima 5051 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
5 | 2, 3, 4 | 3eqtr4i 2642 | 1 ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ran crn 5039 ↾ cres 5040 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: isarep2 5892 f1imacnv 6066 foimacnv 6067 dffv2 6181 islindf4 19996 qtopres 21311 |
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