Theorem rescnvcnv 5250

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 5241	. . 3 |- `' `' A = ( A |` _V )
2	1	reseq1i 5059	. 2 |- ( `' `' A |` B ) = ( ( A |` _V ) |` B )
3		resres 5075	. 2 |- ( ( A |` _V ) |` B ) = ( A |` ( _V i^i B ) )
4		ssv 3270	. . . 4 |- B C_ _V
5		sseqin2 3450	. . . 4 |- ( B C_ _V <-> ( _V i^i B ) = B )
6	4, 5	mpbi 145	. . 3 |- ( _V i^i B ) = B
7	6	reseq2i 5060	. 2 |- ( A |` ( _V i^i B ) ) = ( A |` B )
8	2, 3, 7	3eqtri 2263	1 |- ( `' `' A |` B ) = ( A |` B )

This proof depends on syntax axioms:   = wceq 1402   _Vcvv 2821   i^i cin 3219   C_ wss 3220   `'ccnv 4773   |` cres 4776

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-res 4786

This theorem is used by:  cnvcnvres  5251  imacnvcnv  5252  resdm2  5278  resdmres  5279  coires1  5305  cocnvres  5312  f1oresrab  5873