Theorem cocnvres 5063

Description: Restricting a relation and a converse relation when they are composed together (Contributed by BJ, 10-Jul-2022.)

Assertion

Ref	Expression
cocnvres	|- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )

Proof of Theorem cocnvres

Step	Hyp	Ref	Expression
1		resss 4843	. . . 4 |- ( S |` dom R ) C_ S
2		dmss 4738	. . . 4 |- ( ( S |` dom R ) C_ S -> dom ( S |` dom R ) C_ dom S )
3	1, 2	ax-mp 5	. . 3 |- dom ( S |` dom R ) C_ dom S
4		cores2 5051	. . 3 |- ( dom ( S |` dom R ) C_ dom S -> ( ( S |` dom R ) o. `' ( `' `' R |` dom S ) ) = ( ( S |` dom R ) o. `' R ) )
5	3, 4	ax-mp 5	. 2 |- ( ( S |` dom R ) o. `' ( `' `' R |` dom S ) ) = ( ( S |` dom R ) o. `' R )
6		rescnvcnv 5001	. . . 4 |- ( `' `' R |` dom S ) = ( R |` dom S )
7	6	cnveqi 4714	. . 3 |- `' ( `' `' R |` dom S ) = `' ( R |` dom S )
8	7	coeq2i 4699	. 2 |- ( ( S |` dom R ) o. `' ( `' `' R |` dom S ) ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )
9		dfdm4 4731	. . . 4 |- dom R = ran `' R
10	9	eqimss2i 3154	. . 3 |- ran `' R C_ dom R
11		cores 5042	. . 3 |- ( ran `' R C_ dom R -> ( ( S |` dom R ) o. `' R ) = ( S o. `' R ) )
12	10, 11	ax-mp 5	. 2 |- ( ( S |` dom R ) o. `' R ) = ( S o. `' R )
13	5, 8, 12	3eqtr3ri 2169	1 |- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )

Syntax hints:   = wceq 1331   C_ wss 3071   `'ccnv 4538   dom cdm 4539   ran crn 4540   |` cres 4541   o. ccom 4543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551

This theorem is referenced by:  cocnvss  5064