Theorem cnvcnvres 5107

Description: The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)

Assertion

Ref	Expression
cnvcnvres	|- `' `' ( A |` B ) = ( `' `' A |` B )

Proof of Theorem cnvcnvres

Step	Hyp	Ref	Expression
1		relres 4950	. . 3 |- Rel ( A |` B )
2		dfrel2 5094	. . 3 |- ( Rel ( A |` B ) <-> `' `' ( A |` B ) = ( A |` B ) )
3	1, 2	mpbi 145	. 2 |- `' `' ( A |` B ) = ( A |` B )
4		rescnvcnv 5106	. 2 |- ( `' `' A |` B ) = ( A |` B )
5	3, 4	eqtr4i 2213	1 |- `' `' ( A |` B ) = ( `' `' A |` B )

Syntax hints:   = wceq 1364   `'ccnv 4640   |` cres 4643   Rel wrel 4646

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 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648  df-cnv 4649  df-res 4653

This theorem is referenced by: (None)