Theorem rescnvcnv 5075

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 5066	. . 3 |- `' `' A = ( A |` _V )
2	1	reseq1i 4888	. 2 |- ( `' `' A |` B ) = ( ( A |` _V ) |` B )
3		resres 4904	. 2 |- ( ( A |` _V ) |` B ) = ( A |` ( _V i^i B ) )
4		ssv 3170	. . . 4 |- B C_ _V
5		sseqin2 3347	. . . 4 |- ( B C_ _V <-> ( _V i^i B ) = B )
6	4, 5	mpbi 144	. . 3 |- ( _V i^i B ) = B
7	6	reseq2i 4889	. 2 |- ( A |` ( _V i^i B ) ) = ( A |` B )
8	2, 3, 7	3eqtri 2196	1 |- ( `' `' A |` B ) = ( A |` B )

Syntax hints:   = wceq 1349   _Vcvv 2731   i^i cin 3121   C_ wss 3122   `'ccnv 4611   |` cres 4614

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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-v 2733  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-br 3991  df-opab 4052  df-xp 4618  df-rel 4619  df-cnv 4620  df-res 4624

This theorem is referenced by:  cnvcnvres  5076  imacnvcnv  5077  resdm2  5103  resdmres  5104  coires1  5130  cocnvres  5137  f1oresrab  5665