Theorem cocnvres 5135

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 4915	. . . 4 |- ( S |` dom R ) C_ S
2		dmss 4810	. . . 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 5123	. . 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 5073	. . . 4 |- ( `' `' R |` dom S ) = ( R |` dom S )
7	6	cnveqi 4786	. . 3 |- `' ( `' `' R |` dom S ) = `' ( R |` dom S )
8	7	coeq2i 4771	. 2 |- ( ( S |` dom R ) o. `' ( `' `' R |` dom S ) ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )
9		dfdm4 4803	. . . 4 |- dom R = ran `' R
10	9	eqimss2i 3204	. . 3 |- ran `' R C_ dom R
11		cores 5114	. . 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 2200	1 |- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )

Syntax hints:   = wceq 1348   C_ wss 3121   `'ccnv 4610   dom cdm 4611   ran crn 4612   |` cres 4613   o. ccom 4615

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623

This theorem is referenced by:  cocnvss  5136