Theorem cocnvres 5190

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 4966	. . . 4 |- ( S |` dom R ) C_ S
2		dmss 4861	. . . 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 5178	. . 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 5128	. . . 4 |- ( `' `' R |` dom S ) = ( R |` dom S )
7	6	cnveqi 4837	. . 3 |- `' ( `' `' R |` dom S ) = `' ( R |` dom S )
8	7	coeq2i 4822	. 2 |- ( ( S |` dom R ) o. `' ( `' `' R |` dom S ) ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )
9		dfdm4 4854	. . . 4 |- dom R = ran `' R
10	9	eqimss2i 3236	. . 3 |- ran `' R C_ dom R
11		cores 5169	. . 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 2223	1 |- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )

Syntax hints:   = wceq 1364   C_ wss 3153   `'ccnv 4658   dom cdm 4659   ran crn 4660   |` cres 4661   o. ccom 4663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671

This theorem is referenced by:  cocnvss  5191