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Theorem resima 4969
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 4968 . . 3 ((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)
21rneqi 4884 . 2 ran ((𝐴𝐵) ↾ 𝐵) = ran (𝐴𝐵)
3 df-ima 4668 . 2 ((𝐴𝐵) “ 𝐵) = ran ((𝐴𝐵) ↾ 𝐵)
4 df-ima 4668 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2224 1 ((𝐴𝐵) “ 𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1364  ran crn 4656  cres 4657  cima 4658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4661  df-rel 4662  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668
This theorem is referenced by:  isarep2  5333  f1imacnv  5509  foimacnv  5510  djudm  7154  suplocexprlemell  7763  elq  9677  qnnen  12575
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