Metamath Proof Explorer

Theorem resima2

Description: Image under a restricted class. (Contributed by FL, 31-Aug-2009) (Proof shortened by JJ, 25-Aug-2021)

Ref Expression
Assertion resima2 
|- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )

Proof

Step Hyp Ref Expression
1 sseqin2 
 |-  ( B C_ C <-> ( C i^i B ) = B )
2 reseq2 
 |-  ( ( C i^i B ) = B -> ( A |` ( C i^i B ) ) = ( A |` B ) )
3 1 2 sylbi 
 |-  ( B C_ C -> ( A |` ( C i^i B ) ) = ( A |` B ) )
4 3 rneqd 
 |-  ( B C_ C -> ran ( A |` ( C i^i B ) ) = ran ( A |` B ) )
5 df-ima 
 |-  ( ( A |` C ) " B ) = ran ( ( A |` C ) |` B )
6 resres 
 |-  ( ( A |` C ) |` B ) = ( A |` ( C i^i B ) )
7 6 rneqi 
 |-  ran ( ( A |` C ) |` B ) = ran ( A |` ( C i^i B ) )
8 5 7 eqtri 
 |-  ( ( A |` C ) " B ) = ran ( A |` ( C i^i B ) )
9 df-ima 
 |-  ( A " B ) = ran ( A |` B )
10 4 8 9 3eqtr4g 
 |-  ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )