Theorem cnvcnv 5240

Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)

Assertion

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

Proof of Theorem cnvcnv

Step	Hyp	Ref	Expression
1		relcnv 5165	. . . . 5 |- Rel `' `' A
2		df-rel 4781	. . . . 5 |- ( Rel `' `' A <-> `' `' A C_ ( _V X. _V ) )
3	1, 2	mpbi 145	. . . 4 |- `' `' A C_ ( _V X. _V )
4		relxp 4884	. . . . 5 |- Rel ( _V X. _V )
5		dfrel2 5238	. . . . 5 |- ( Rel ( _V X. _V ) <-> `' `' ( _V X. _V ) = ( _V X. _V ) )
6	4, 5	mpbi 145	. . . 4 |- `' `' ( _V X. _V ) = ( _V X. _V )
7	3, 6	sseqtrri 3283	. . 3 |- `' `' A C_ `' `' ( _V X. _V )
8		dfss 3234	. . 3 |- ( `' `' A C_ `' `' ( _V X. _V ) <-> `' `' A = ( `' `' A i^i `' `' ( _V X. _V ) ) )
9	7, 8	mpbi 145	. 2 |- `' `' A = ( `' `' A i^i `' `' ( _V X. _V ) )
10		cnvin 5195	. 2 |- `' ( `' A i^i `' ( _V X. _V ) ) = ( `' `' A i^i `' `' ( _V X. _V ) )
11		cnvin 5195	. . . 4 |- `' ( A i^i ( _V X. _V ) ) = ( `' A i^i `' ( _V X. _V ) )
12	11	cnveqi 4955	. . 3 |- `' `' ( A i^i ( _V X. _V ) ) = `' ( `' A i^i `' ( _V X. _V ) )
13		inss2 3452	. . . . 5 |- ( A i^i ( _V X. _V ) ) C_ ( _V X. _V )
14		df-rel 4781	. . . . 5 |- ( Rel ( A i^i ( _V X. _V ) ) <-> ( A i^i ( _V X. _V ) ) C_ ( _V X. _V ) )
15	13, 14	mpbir 146	. . . 4 |- Rel ( A i^i ( _V X. _V ) )
16		dfrel2 5238	. . . 4 |- ( Rel ( A i^i ( _V X. _V ) ) <-> `' `' ( A i^i ( _V X. _V ) ) = ( A i^i ( _V X. _V ) ) )
17	15, 16	mpbi 145	. . 3 |- `' `' ( A i^i ( _V X. _V ) ) = ( A i^i ( _V X. _V ) )
18	12, 17	eqtr3i 2261	. 2 |- `' ( `' A i^i `' ( _V X. _V ) ) = ( A i^i ( _V X. _V ) )
19	9, 10, 18	3eqtr2i 2265	1 |- `' `' A = ( A i^i ( _V X. _V ) )

This proof depends on syntax axioms:   = wceq 1402   _Vcvv 2821   i^i cin 3219   C_ wss 3220   X. cxp 4772   `'ccnv 4773   Rel wrel 4779

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

This theorem is used by:  cnvcnv2  5241  cnvcnvss  5242  structcnvcnv  13368  strslfv2d  13395