Theorem rescnvcnv 5402

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 5392	. . 3 |- `' `' A = ( A |` _V )
2	1	reseq1i 5207	. 2 |- ( `' `' A |` B ) = ( ( A |` _V ) |` B )
3		resres 5224	. 2 |- ( ( A |` _V ) |` B ) = ( A |` ( _V i^i B ) )
4		ssv 3477	. . . 4 |- B C_ _V
5		sseqin2 3670	. . . 4 |- ( B C_ _V <-> ( _V i^i B ) = B )
6	4, 5	mpbi 208	. . 3 |- ( _V i^i B ) = B
7	6	reseq2i 5208	. 2 |- ( A |` ( _V i^i B ) ) = ( A |` B )
8	2, 3, 7	3eqtri 2484	1 |- ( `' `' A |` B ) = ( A |` B )

Syntax hints:   = wceq 1370   _Vcvv 3071   i^i cin 3428   C_ wss 3429   `'ccnv 4940   |` cres 4943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-cnv 4949  df-res 4953

This theorem is referenced by:  cnvcnvres  5403  imacnvcnv  5404  resdm2  5429  resdmres  5430  coires1  5456  f1oresrab  5977