Theorem rescnvcnv 5106

Description: The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

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

Proof of Theorem rescnvcnv

Step	Hyp	Ref	Expression
1		cnvcnv2 5097	. . 3 |- `' `' A = ( A |` _V )
2	1	reseq1i 4918	. 2 |- ( `' `' A |` B ) = ( ( A |` _V ) |` B )
3		resres 4934	. 2 |- ( ( A |` _V ) |` B ) = ( A |` ( _V i^i B ) )
4		ssv 3192	. . . 4 |- B C_ _V
5		sseqin2 3369	. . . 4 |- ( B C_ _V <-> ( _V i^i B ) = B )
6	4, 5	mpbi 145	. . 3 |- ( _V i^i B ) = B
7	6	reseq2i 4919	. 2 |- ( A |` ( _V i^i B ) ) = ( A |` B )
8	2, 3, 7	3eqtri 2214	1 |- ( `' `' A |` B ) = ( A |` B )

Syntax hints:   = wceq 1364   _Vcvv 2752   i^i cin 3143   C_ wss 3144   `'ccnv 4640   |` cres 4643

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:  cnvcnvres  5107  imacnvcnv  5108  resdm2  5134  resdmres  5135  coires1  5161  cocnvres  5168  f1oresrab  5697