Theorem cnvcnv 4359

Description: The double converse of a class strips out all elements that are not ordered pairs.

Assertion

Ref	Expression
cnvcnv	|- `'`'A = (A i^i (_V X. _V))

Proof of Theorem cnvcnv

Step	Hyp	Ref	Expression
1		cnvin 4324	. . 3 |- `'(A i^i (_V X. _V)) = (`'A i^i `'(_V X. _V))
2	1	cnveqi 4136	. 2 |- `'`'(A i^i (_V X. _V)) = `'(`'A i^i `'(_V X. _V))
3		inss2 2813	. . . 4 |- (A i^i (_V X. _V)) C_ (_V X. _V)
4		df-rel 4001	. . . 4 |- (Rel (A i^i (_V X. _V)) <-> (A i^i (_V X. _V)) C_ (_V X. _V))
5	3, 4	mpbir 207	. . 3 |- Rel (A i^i (_V X. _V))
6		dfrel2 4358	. . 3 |- (Rel (A i^i (_V X. _V)) <-> `'`'(A i^i (_V X. _V)) = (A i^i (_V X. _V)))
7	5, 6	mpbi 206	. 2 |- `'`'(A i^i (_V X. _V)) = (A i^i (_V X. _V))
8		cnvin 4324	. . 3 |- `'(`'A i^i `'(_V X. _V)) = (`'`'A i^i `'`'(_V X. _V))
9		relcnv 4301	. . . . . 6 |- Rel `'`'A
10		df-rel 4001	. . . . . 6 |- (Rel `'`'A <-> `'`'A C_ (_V X. _V))
11	9, 10	mpbi 206	. . . . 5 |- `'`'A C_ (_V X. _V)
12		relxp 4088	. . . . . 6 |- Rel (_V X. _V)
13		dfrel2 4358	. . . . . 6 |- (Rel (_V X. _V) <-> `'`'(_V X. _V) = (_V X. _V))
14	12, 13	mpbi 206	. . . . 5 |- `'`'(_V X. _V) = (_V X. _V)
15	11, 14	sseqtr4i 2650	. . . 4 |- `'`'A C_ `'`'(_V X. _V)
16		dfss 2606	. . . 4 |- (`'`'A C_ `'`'(_V X. _V) <-> `'`'A = (`'`'A i^i `'`'(_V X. _V)))
17	15, 16	mpbi 206	. . 3 |- `'`'A = (`'`'A i^i `'`'(_V X. _V))
18	8, 17	eqtr4i 1911	. 2 |- `'(`'A i^i `'(_V X. _V)) = `'`'A
19	2, 7, 18	3eqtr3ri 1920	1 |- `'`'A = (A i^i (_V X. _V))

Syntax hints:   = wceq 1298  _Vcvv 2292   i^i cin 2592   C_ wss 2593   X. cxp 3984  `'ccnv 3985  Rel wrel 3991

This theorem is referenced by:  cnvcnv2 4360  cnvcnvss 4361  dmsnn0OLD 4363  rescnvcnvOLD 4386

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

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