Theorem resima 4808

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 4807	. . 3 |- ( ( A |` B ) |` B ) = ( A |` B )
2	1	rneqi 4725	. 2 |- ran ( ( A |` B ) |` B ) = ran ( A |` B )
3		df-ima 4510	. 2 |- ( ( A |` B ) " B ) = ran ( ( A |` B ) |` B )
4		df-ima 4510	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2143	1 |- ( ( A |` B ) " B ) = ( A " B )

Syntax hints:   = wceq 1312   ran crn 4498   |` cres 4499   "cima 4500

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-rel 4504  df-cnv 4505  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510

This theorem is referenced by:  isarep2  5166  f1imacnv  5338  foimacnv  5339  djudm  6940  elq  9310  qnnen  11783