Theorem resima 4991

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 4990	. . 3 |- ( ( A |` B ) |` B ) = ( A |` B )
2	1	rneqi 4905	. 2 |- ran ( ( A |` B ) |` B ) = ran ( A |` B )
3		df-ima 4687	. 2 |- ( ( A |` B ) " B ) = ran ( ( A |` B ) |` B )
4		df-ima 4687	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2235	1 |- ( ( A |` B ) " B ) = ( A " B )

Syntax hints:   = wceq 1372   ran crn 4675   |` cres 4676   "cima 4677

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687

This theorem is referenced by:  isarep2  5360  f1imacnv  5538  foimacnv  5539  djudm  7206  suplocexprlemell  7825  elq  9742  qnnen  12773