Theorem resima 4901

Description: A restriction to an image. (Contributed by NM, 29-Sep-2004.)

Assertion

Ref	Expression
resima	|- ( ( A |` B ) " B ) = ( A " B )

Proof of Theorem resima

Step	Hyp	Ref	Expression
1		residm 4900	. . 3 |- ( ( A |` B ) |` B ) = ( A |` B )
2	1	rneqi 4816	. 2 |- ran ( ( A |` B ) |` B ) = ran ( A |` B )
3		df-ima 4601	. 2 |- ( ( A |` B ) " B ) = ran ( ( A |` B ) |` B )
4		df-ima 4601	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2188	1 |- ( ( A |` B ) " B ) = ( A " B )

Syntax hints:   = wceq 1335   ran crn 4589   |` cres 4590   "cima 4591

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594  df-rel 4595  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601

This theorem is referenced by:  isarep2  5259  f1imacnv  5433  foimacnv  5434  djudm  7051  suplocexprlemell  7635  elq  9537  qnnen  12230