Theorem cnvcnv 3478

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 3448	. . 3 |- `'(A i^i (V X. V)) = (`'A i^i `'(V X. V))
2		cnveq 3287	. . 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 2227	. . . 4 |- (A i^i (V X. V)) (_ (V X. V)
5		df-rel 3180	. . . 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 3477	. . 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 3448	. . 3 |- `'(`'A i^i `'(V X. V)) = (`'`'A i^i `'`'(V X. V))
10		relcnv 3427	. . . . . 6 |- Rel `'`'A
11		df-rel 3180	. . . . . 6 |- (Rel `'`'A <-> `'`'A (_ (V X. V))
12	10, 11	mpbi 189	. . . . 5 |- `'`'A (_ (V X. V)
13		relxp 3250	. . . . . 6 |- Rel (V X. V)
14		dfrel2 3477	. . . . . 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 2090	. . . 4 |- `'`'A (_ `'`'(V X. V)
17		dfss 2050	. . . 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 1495	. 2 |- `'(`'A i^i `'(V X. V)) = `'`'A
20	3, 8, 19	3eqtr3r 1501	1 |- `'`'A = (A i^i (V X. V))

Syntax hints:   = wceq 954  Vcvv 1807   i^i cin 2042   (_ wss 2043   X. cxp 3163  `'ccnv 3164  Rel wrel 3170

This theorem is referenced by:  cnvcnv2 3479  cnvcnvss 3480  rescnvcnv 3485

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-cnv 3181

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