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Mirrors > Home > ILE Home > Th. List > rnresi | GIF version |
Description: The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.) |
Ref | Expression |
---|---|
rnresi | ⊢ ran ( I ↾ 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 4547 | . 2 ⊢ ( I “ 𝐴) = ran ( I ↾ 𝐴) | |
2 | imai 4890 | . 2 ⊢ ( I “ 𝐴) = 𝐴 | |
3 | 1, 2 | eqtr3i 2160 | 1 ⊢ ran ( I ↾ 𝐴) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 I cid 4205 ran crn 4535 ↾ cres 4536 “ cima 4537 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 |
This theorem is referenced by: resiima 4892 iordsmo 6187 restid2 12118 |
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