Theorem rescnvcnv 4385

Description: The restriction of the double converse of a class. (The proof was 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 4360	. . 3 |- `'`'A = (A |` _V)
2		reseq1 4218	. . 3 |- (`'`'A = (A |` _V) -> (`'`'A |` B) = ((A |` _V) |` B))
3	1, 2	ax-mp 7	. 2 |- (`'`'A |` B) = ((A |` _V) |` B)
4		resres 4228	. 2 |- ((A |` _V) |` B) = (A |` (_V i^i B))
5		ssv 2636	. . . 4 |- B C_ _V
6		sseqin2 2811	. . . 4 |- (B C_ _V <-> (_V i^i B) = B)
7	5, 6	mpbi 206	. . 3 |- (_V i^i B) = B
8		reseq2 4219	. . 3 |- ((_V i^i B) = B -> (A |` (_V i^i B)) = (A |` B))
9	7, 8	ax-mp 7	. 2 |- (A |` (_V i^i B)) = (A |` B)
10	3, 4, 9	3eqtri 1912	1 |- (`'`'A |` B) = (A |` B)

Syntax hints:   = wceq 1298  _Vcvv 2292   i^i cin 2592   C_ wss 2593  `'ccnv 3985   |` cres 3988

This theorem is referenced by:  cnvcnvres 4387  imacnvcnv 4388  resdm2 4389  resdmres 4390  domrancur1c 14550

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-res 4006

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