Theorem cnvcnv 5118

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 5043	. . . . 5 ⊢ Rel ◡◡𝐴
2		df-rel 4666	. . . . 5 ⊢ (Rel ◡◡𝐴 ↔ ◡◡𝐴 ⊆ (V × V))
3	1, 2	mpbi 145	. . . 4 ⊢ ◡◡𝐴 ⊆ (V × V)
4		relxp 4768	. . . . 5 ⊢ Rel (V × V)
5		dfrel2 5116	. . . . 5 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V))
6	4, 5	mpbi 145	. . . 4 ⊢ ◡◡(V × V) = (V × V)
7	3, 6	sseqtrri 3214	. . 3 ⊢ ◡◡𝐴 ⊆ ◡◡(V × V)
8		dfss 3167	. . 3 ⊢ (◡◡𝐴 ⊆ ◡◡(V × V) ↔ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V)))
9	7, 8	mpbi 145	. 2 ⊢ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V))
10		cnvin 5073	. 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (◡◡𝐴 ∩ ◡◡(V × V))
11		cnvin 5073	. . . 4 ⊢ ◡(𝐴 ∩ (V × V)) = (◡𝐴 ∩ ◡(V × V))
12	11	cnveqi 4837	. . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = ◡(◡𝐴 ∩ ◡(V × V))
13		inss2 3380	. . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V)
14		df-rel 4666	. . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
15	13, 14	mpbir 146	. . . 4 ⊢ Rel (𝐴 ∩ (V × V))
16		dfrel2 5116	. . . 4 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
17	15, 16	mpbi 145	. . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
18	12, 17	eqtr3i 2216	. 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (𝐴 ∩ (V × V))
19	9, 10, 18	3eqtr2i 2220	1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))

Syntax hints:   = wceq 1364  Vcvv 2760   ∩ cin 3152   ⊆ wss 3153   × cxp 4657  ^◡ccnv 4658  Rel wrel 4664

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667

This theorem is referenced by:  cnvcnv2  5119  cnvcnvss  5120  structcnvcnv  12634  strslfv2d  12661