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Metamath Proof Explorer

Theorem cores2

Description: Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006)

Ref Expression
Assertion cores2 
|- ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) )

Proof

Step Hyp Ref Expression
1 dfdm4 
 |-  dom A = ran `' A
2 1 sseq1i 
 |-  ( dom A C_ C <-> ran `' A C_ C )
3 cores 
 |-  ( ran `' A C_ C -> ( ( `' B |` C ) o. `' A ) = ( `' B o. `' A ) )
4 2 3 sylbi 
 |-  ( dom A C_ C -> ( ( `' B |` C ) o. `' A ) = ( `' B o. `' A ) )
5 cnvco 
 |-  `' ( A o. `' ( `' B |` C ) ) = ( `' `' ( `' B |` C ) o. `' A )
6 cocnvcnv1 
 |-  ( `' `' ( `' B |` C ) o. `' A ) = ( ( `' B |` C ) o. `' A )
7 5 6 eqtri 
 |-  `' ( A o. `' ( `' B |` C ) ) = ( ( `' B |` C ) o. `' A )
8 cnvco 
 |-  `' ( A o. B ) = ( `' B o. `' A )
9 4 7 8 3eqtr4g 
 |-  ( dom A C_ C -> `' ( A o. `' ( `' B |` C ) ) = `' ( A o. B ) )
10 9 cnveqd 
 |-  ( dom A C_ C -> `' `' ( A o. `' ( `' B |` C ) ) = `' `' ( A o. B ) )
11 relco 
 |-  Rel ( A o. `' ( `' B |` C ) )
12 dfrel2 
 |-  ( Rel ( A o. `' ( `' B |` C ) ) <-> `' `' ( A o. `' ( `' B |` C ) ) = ( A o. `' ( `' B |` C ) ) )
13 11 12 mpbi 
 |-  `' `' ( A o. `' ( `' B |` C ) ) = ( A o. `' ( `' B |` C ) )
14 relco 
 |-  Rel ( A o. B )
15 dfrel2 
 |-  ( Rel ( A o. B ) <-> `' `' ( A o. B ) = ( A o. B ) )
16 14 15 mpbi 
 |-  `' `' ( A o. B ) = ( A o. B )
17 10 13 16 3eqtr3g 
 |-  ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) )