Theorem cocnvres 4955

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 4737	. . . 4 |- ( S |` dom R ) C_ S
2		dmss 4635	. . . 4 |- ( ( S |` dom R ) C_ S -> dom ( S |` dom R ) C_ dom S )
3	1, 2	ax-mp 7	. . 3 |- dom ( S |` dom R ) C_ dom S
4		cores2 4943	. . 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 7	. 2 |- ( ( S |` dom R ) o. `' ( `' `' R |` dom S ) ) = ( ( S |` dom R ) o. `' R )
6		rescnvcnv 4893	. . . 4 |- ( `' `' R |` dom S ) = ( R |` dom S )
7	6	cnveqi 4611	. . 3 |- `' ( `' `' R |` dom S ) = `' ( R |` dom S )
8	7	coeq2i 4596	. 2 |- ( ( S |` dom R ) o. `' ( `' `' R |` dom S ) ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )
9		dfdm4 4628	. . . 4 |- dom R = ran `' R
10	9	eqimss2i 3081	. . 3 |- ran `' R C_ dom R
11		cores 4934	. . 3 |- ( ran `' R C_ dom R -> ( ( S |` dom R ) o. `' R ) = ( S o. `' R ) )
12	10, 11	ax-mp 7	. 2 |- ( ( S |` dom R ) o. `' R ) = ( S o. `' R )
13	5, 8, 12	3eqtr3ri 2117	1 |- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )

Syntax hints:   = wceq 1289   C_ wss 2999   `'ccnv 4437   dom cdm 4438   ran crn 4439   |` cres 4440   o. ccom 4442

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450

This theorem is referenced by:  cocnvss  4956