Theorem cnvcnv 3492

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 3462	. . 3 |- `'(A i^i (V X. V)) = (`'A i^i `'(V X. V))
2		cnveq 3298	. . 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 7	. 2 |- `'`'(A i^i (V X. V)) = `'(`'A i^i `'(V X. V))
4		inss2 2234	. . . 4 |- (A i^i (V X. V)) (_ (V X. V)
5		df-rel 3191	. . . 4 |- (Rel (A i^i (V X. V)) <-> (A i^i (V X. V)) (_ (V X. V))
6	4, 5	mpbir 190	. . 3 |- Rel (A i^i (V X. V))
7		dfrel2 3491	. . 3 |- (Rel (A i^i (V X. V)) <-> `'`'(A i^i (V X. V)) = (A i^i (V X. V)))
8	6, 7	mpbi 189	. 2 |- `'`'(A i^i (V X. V)) = (A i^i (V X. V))
9		cnvin 3462	. . 3 |- `'(`'A i^i `'(V X. V)) = (`'`'A i^i `'`'(V X. V))
10		relcnv 3441	. . . . . 6 |- Rel `'`'A
11		df-rel 3191	. . . . . 6 |- (Rel `'`'A <-> `'`'A (_ (V X. V))
12	10, 11	mpbi 189	. . . . 5 |- `'`'A (_ (V X. V)
13		relxp 3261	. . . . . 6 |- Rel (V X. V)
14		dfrel2 3491	. . . . . 6 |- (Rel (V X. V) <-> `'`'(V X. V) = (V X. V))
15	13, 14	mpbi 189	. . . . 5 |- `'`'(V X. V) = (V X. V)
16	12, 15	sseqtr4 2097	. . . 4 |- `'`'A (_ `'`'(V X. V)
17		dfss 2057	. . . 4 |- (`'`'A (_ `'`'(V X. V) <-> `'`'A = (`'`'A i^i `'`'(V X. V)))
18	16, 17	mpbi 189	. . 3 |- `'`'A = (`'`'A i^i `'`'(V X. V))
19	9, 18	eqtr4 1501	. 2 |- `'(`'A i^i `'(V X. V)) = `'`'A
20	3, 8, 19	3eqtr3r 1507	1 |- `'`'A = (A i^i (V X. V))

Syntax hints:   = wceq 958  Vcvv 1814   i^i cin 2049   (_ wss 2050   X. cxp 3174  `'ccnv 3175  Rel wrel 3181

This theorem is referenced by:  cnvcnv2 3493  cnvcnvss 3494  rescnvcnv 3499

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192

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