Theorem cnvcnv 5031

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 ∩ (V × V))

Proof of Theorem cnvcnv

Step	Hyp	Ref	Expression
1		relcnv 4957	. . . . 5 ⊢ Rel ◡◡𝐴
2		df-rel 4586	. . . . 5 ⊢ (Rel ◡◡𝐴 ↔ ◡◡𝐴 ⊆ (V × V))
3	1, 2	mpbi 144	. . . 4 ⊢ ◡◡𝐴 ⊆ (V × V)
4		relxp 4688	. . . . 5 ⊢ Rel (V × V)
5		dfrel2 5029	. . . . 5 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V))
6	4, 5	mpbi 144	. . . 4 ⊢ ◡◡(V × V) = (V × V)
7	3, 6	sseqtrri 3159	. . 3 ⊢ ◡◡𝐴 ⊆ ◡◡(V × V)
8		dfss 3112	. . 3 ⊢ (◡◡𝐴 ⊆ ◡◡(V × V) ↔ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V)))
9	7, 8	mpbi 144	. 2 ⊢ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V))
10		cnvin 4986	. 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (◡◡𝐴 ∩ ◡◡(V × V))
11		cnvin 4986	. . . 4 ⊢ ◡(𝐴 ∩ (V × V)) = (◡𝐴 ∩ ◡(V × V))
12	11	cnveqi 4754	. . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = ◡(◡𝐴 ∩ ◡(V × V))
13		inss2 3324	. . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V)
14		df-rel 4586	. . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
15	13, 14	mpbir 145	. . . 4 ⊢ Rel (𝐴 ∩ (V × V))
16		dfrel2 5029	. . . 4 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
17	15, 16	mpbi 144	. . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
18	12, 17	eqtr3i 2177	. 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (𝐴 ∩ (V × V))
19	9, 10, 18	3eqtr2i 2181	1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))

Syntax hints:   = wceq 1332  Vcvv 2709   ∩ cin 3097   ⊆ wss 3098   × cxp 4577  ^◡ccnv 4578  Rel wrel 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-xp 4585  df-rel 4586  df-cnv 4587

This theorem is referenced by:  cnvcnv2  5032  cnvcnvss  5033  structcnvcnv  12153  strslfv2d  12179