Theorem imadmres 5627

Description: The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
imadmres	|- ( A " dom ( A |` B ) ) = ( A " B )

Proof of Theorem imadmres

Step	Hyp	Ref	Expression
1		resdmres 5625	. . 3 |- ( A |` dom ( A |` B ) ) = ( A |` B )
2	1	rneqi 5352	. 2 |- ran ( A |` dom ( A |` B ) ) = ran ( A |` B )
3		df-ima 5127	. 2 |- ( A " dom ( A |` B ) ) = ran ( A |` dom ( A |` B ) )
4		df-ima 5127	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2654	1 |- ( A " dom ( A |` B ) ) = ( A " B )

Syntax hints:   = wceq 1483   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  ssimaex  6263  fnwelem  7292  imafi  8259  r0weon  8835  limsupgle  14208  kqdisj  21535