Theorem cnvcnv 5154

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 5079	. . . . 5 |- Rel `' `' A
2		df-rel 4700	. . . . 5 |- ( Rel `' `' A <-> `' `' A C_ ( _V X. _V ) )
3	1, 2	mpbi 145	. . . 4 |- `' `' A C_ ( _V X. _V )
4		relxp 4802	. . . . 5 |- Rel ( _V X. _V )
5		dfrel2 5152	. . . . 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 3236	. . 3 |- `' `' A C_ `' `' ( _V X. _V )
8		dfss 3188	. . 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 5109	. 2 |- `' ( `' A i^i `' ( _V X. _V ) ) = ( `' `' A i^i `' `' ( _V X. _V ) )
11		cnvin 5109	. . . 4 |- `' ( A i^i ( _V X. _V ) ) = ( `' A i^i `' ( _V X. _V ) )
12	11	cnveqi 4871	. . 3 |- `' `' ( A i^i ( _V X. _V ) ) = `' ( `' A i^i `' ( _V X. _V ) )
13		inss2 3402	. . . . 5 |- ( A i^i ( _V X. _V ) ) C_ ( _V X. _V )
14		df-rel 4700	. . . . 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 5152	. . . 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 2230	. 2 |- `' ( `' A i^i `' ( _V X. _V ) ) = ( A i^i ( _V X. _V ) )
19	9, 10, 18	3eqtr2i 2234	1 |- `' `' A = ( A i^i ( _V X. _V ) )

Syntax hints:   = wceq 1373   _Vcvv 2776   i^i cin 3173   C_ wss 3174   X. cxp 4691   `'ccnv 4692   Rel wrel 4698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701

This theorem is referenced by:  cnvcnv2  5155  cnvcnvss  5156  structcnvcnv  12963  strslfv2d  12990