Theorem cnvcnv 2661

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 2643	. . 3 |- `'(A i^i (V X. V)) = (`'A i^i `'(V X. V))
2		cnveq 2513	. . 3 |- (`'(A i^i (V X. V)) = (`'A i^i `'(V X. V)) -> `'`'(A i^i (V X. V)) = `'(`'A i^i `'(V X. V)))
3	1, 2	ax-mp 6	. 2 |- `'`'(A i^i (V X. V)) = `'(`'A i^i `'(V X. V))
4		inss2 1658	. . . 4 |- (A i^i (V X. V)) (_ (V X. V)
5		df-rel 2425	. . . 4 |- (Rel (A i^i (V X. V)) <-> (A i^i (V X. V)) (_ (V X. V))
6	4, 5	mpbir 165	. . 3 |- Rel (A i^i (V X. V))
7		dfrel2 2660	. . 3 |- (Rel (A i^i (V X. V)) <-> `'`'(A i^i (V X. V)) = (A i^i (V X. V)))
8	6, 7	mpbi 164	. 2 |- `'`'(A i^i (V X. V)) = (A i^i (V X. V))
9		cnvin 2643	. . 3 |- `'(`'A i^i `'(V X. V)) = (`'`'A i^i `'`'(V X. V))
10		relcnv 2624	. . . . . 6 |- Rel `'`'A
11		df-rel 2425	. . . . . 6 |- (Rel `'`'A <-> `'`'A (_ (V X. V))
12	10, 11	mpbi 164	. . . . 5 |- `'`'A (_ (V X. V)
13		relxp 2486	. . . . . 6 |- Rel (V X. V)
14		dfrel2 2660	. . . . . 6 |- (Rel (V X. V) <-> `'`'(V X. V) = (V X. V))
15	13, 14	mpbi 164	. . . . 5 |- `'`'(V X. V) = (V X. V)
16	12, 15	sseqtr4 1533	. . . 4 |- `'`'A (_ `'`'(V X. V)
17		dfss 1493	. . . 4 |- (`'`'A (_ `'`'(V X. V) <-> `'`'A = (`'`'A i^i `'`'(V X. V)))
18	16, 17	mpbi 164	. . 3 |- `'`'A = (`'`'A i^i `'`'(V X. V))
19	9, 18	eqtr4 1122	. 2 |- `'(`'A i^i `'(V X. V)) = `'`'A
20	3, 8, 19	3eqtr3r 1125	1 |- `'`'A = (A i^i (V X. V))

Syntax hints:   = wceq 1091  Vcvv 1348   i^i cin 1486   (_ wss 1487   X. cxp 2408  `'ccnv 2409  Rel wrel 2415

This theorem is referenced by:  cnvcnvss 2662

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426

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