Theorem cocnvres 5071

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 4851	. . . 4 |- ( S |` dom R ) C_ S
2		dmss 4746	. . . 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 5059	. . 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 5009	. . . 4 |- ( `' `' R |` dom S ) = ( R |` dom S )
7	6	cnveqi 4722	. . 3 |- `' ( `' `' R |` dom S ) = `' ( R |` dom S )
8	7	coeq2i 4707	. 2 |- ( ( S |` dom R ) o. `' ( `' `' R |` dom S ) ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )
9		dfdm4 4739	. . . 4 |- dom R = ran `' R
10	9	eqimss2i 3159	. . 3 |- ran `' R C_ dom R
11		cores 5050	. . 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 2170	1 |- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )

Syntax hints:   = wceq 1332   C_ wss 3076   `'ccnv 4546   dom cdm 4547   ran crn 4548   |` cres 4549   o. ccom 4551

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559

This theorem is referenced by:  cocnvss  5072