Theorem resima 5098

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 5097	. . 3 |- ( ( A |` B ) |` B ) = ( A |` B )
2	1	rneqi 5012	. 2 |- ran ( ( A |` B ) |` B ) = ran ( A |` B )
3		df-ima 4789	. 2 |- ( ( A |` B ) " B ) = ran ( ( A |` B ) |` B )
4		df-ima 4789	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2269	1 |- ( ( A |` B ) " B ) = ( A " B )

This proof depends on syntax axioms:   = wceq 1402   ran crn 4777   |` cres 4778   "cima 4779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-xp 4782  df-rel 4783  df-cnv 4784  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789

This theorem is used by:  isarep2  5471  f1imacnv  5659  foimacnv  5660  djudm  7451  suplocexprlemell  8086  elq  10038  qnnen  13380