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 ∩ (V × V))

Proof of Theorem cnvcnv

Step	Hyp	Ref	Expression
1		cnvin 2643	. . 3 ⊢ ◡(A ∩ (V × V)) = (◡A ∩ ◡(V × V))
2		cnveq 2513	. . 3 ⊢ (◡(A ∩ (V × V)) = (◡A ∩ ◡(V × V)) → ◡◡(A ∩ (V × V)) = ◡(◡A ∩ ◡(V × V)))
3	1, 2	ax-mp 6	. 2 ⊢ ◡◡(A ∩ (V × V)) = ◡(◡A ∩ ◡(V × V))
4		inss2 1658	. . . 4 ⊢ (A ∩ (V × V)) ⊆ (V × V)
5		df-rel 2425	. . . 4 ⊢ (Rel (A ∩ (V × V)) ↔ (A ∩ (V × V)) ⊆ (V × V))
6	4, 5	mpbir 165	. . 3 ⊢ Rel (A ∩ (V × V))
7		dfrel2 2660	. . 3 ⊢ (Rel (A ∩ (V × V)) ↔ ◡◡(A ∩ (V × V)) = (A ∩ (V × V)))
8	6, 7	mpbi 164	. 2 ⊢ ◡◡(A ∩ (V × V)) = (A ∩ (V × V))
9		cnvin 2643	. . 3 ⊢ ◡(◡A ∩ ◡(V × V)) = (◡◡A ∩ ◡◡(V × V))
10		relcnv 2624	. . . . . 6 ⊢ Rel ◡◡A
11		df-rel 2425	. . . . . 6 ⊢ (Rel ◡◡A ↔ ◡◡A ⊆ (V × V))
12	10, 11	mpbi 164	. . . . 5 ⊢ ◡◡A ⊆ (V × V)
13		relxp 2486	. . . . . 6 ⊢ Rel (V × V)
14		dfrel2 2660	. . . . . 6 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V))
15	13, 14	mpbi 164	. . . . 5 ⊢ ◡◡(V × V) = (V × V)
16	12, 15	sseqtr4 1533	. . . 4 ⊢ ◡◡A ⊆ ◡◡(V × V)
17		dfss 1493	. . . 4 ⊢ (◡◡A ⊆ ◡◡(V × V) ↔ ◡◡A = (◡◡A ∩ ◡◡(V × V)))
18	16, 17	mpbi 164	. . 3 ⊢ ◡◡A = (◡◡A ∩ ◡◡(V × V))
19	9, 18	eqtr4 1122	. 2 ⊢ ◡(◡A ∩ ◡(V × V)) = ◡◡A
20	3, 8, 19	3eqtr3r 1125	1 ⊢ ◡◡A = (A ∩ (V × V))

Syntax hints:   = wceq 1091  Vcvv 1348   ∩ cin 1486   ⊆ wss 1487   × 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

metamath.org