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Theorem resima 4643
Description: A restriction to an image. (Contributed by NM, 29-Sep-2004.)
Assertion
Ref Expression
resima ((𝐴𝐵) “ 𝐵) = (𝐴𝐵)

Proof of Theorem resima
StepHypRef Expression
1 residm 4642 . . 3 ((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)
21rneqi 4562 . 2 ran ((𝐴𝐵) ↾ 𝐵) = ran (𝐴𝐵)
3 df-ima 4358 . 2 ((𝐴𝐵) “ 𝐵) = ran ((𝐴𝐵) ↾ 𝐵)
4 df-ima 4358 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2070 1 ((𝐴𝐵) “ 𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1243  ran crn 4346  cres 4347  cima 4348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358
This theorem is referenced by:  isarep2  4986  f1imacnv  5143  foimacnv  5144  elq  8555
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