Theorem resima 4661

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 4660	. . 3 |- ( ( A |` B ) |` B ) = ( A |` B )
2	1	rneqi 4580	. 2 |- ran ( ( A |` B ) |` B ) = ran ( A |` B )
3		df-ima 4376	. 2 |- ( ( A |` B ) " B ) = ran ( ( A |` B ) |` B )
4		df-ima 4376	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2111	1 |- ( ( A |` B ) " B ) = ( A " B )

Syntax hints:   = wceq 1284   ran crn 4364   |` cres 4365   "cima 4366

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by:  isarep2  5006  f1imacnv  5163  foimacnv  5164  elq  8707