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Mirrors > Home > ILE Home > Th. List > resima | 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 4859 | . . 3 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
2 | 1 | rneqi 4775 | . 2 ⊢ ran ((𝐴 ↾ 𝐵) ↾ 𝐵) = ran (𝐴 ↾ 𝐵) |
3 | df-ima 4560 | . 2 ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = ran ((𝐴 ↾ 𝐵) ↾ 𝐵) | |
4 | df-ima 4560 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
5 | 2, 3, 4 | 3eqtr4i 2171 | 1 ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ran crn 4548 ↾ cres 4549 “ cima 4550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 |
This theorem is referenced by: isarep2 5218 f1imacnv 5392 foimacnv 5393 djudm 6998 suplocexprlemell 7545 elq 9441 qnnen 11980 |
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