Theorem cnvcnv 5122

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 5047	. . . . 5 ⊢ Rel ◡◡𝐴
2		df-rel 4670	. . . . 5 ⊢ (Rel ◡◡𝐴 ↔ ◡◡𝐴 ⊆ (V × V))
3	1, 2	mpbi 145	. . . 4 ⊢ ◡◡𝐴 ⊆ (V × V)
4		relxp 4772	. . . . 5 ⊢ Rel (V × V)
5		dfrel2 5120	. . . . 5 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V))
6	4, 5	mpbi 145	. . . 4 ⊢ ◡◡(V × V) = (V × V)
7	3, 6	sseqtrri 3218	. . 3 ⊢ ◡◡𝐴 ⊆ ◡◡(V × V)
8		dfss 3171	. . 3 ⊢ (◡◡𝐴 ⊆ ◡◡(V × V) ↔ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V)))
9	7, 8	mpbi 145	. 2 ⊢ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V))
10		cnvin 5077	. 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (◡◡𝐴 ∩ ◡◡(V × V))
11		cnvin 5077	. . . 4 ⊢ ◡(𝐴 ∩ (V × V)) = (◡𝐴 ∩ ◡(V × V))
12	11	cnveqi 4841	. . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = ◡(◡𝐴 ∩ ◡(V × V))
13		inss2 3384	. . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V)
14		df-rel 4670	. . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
15	13, 14	mpbir 146	. . . 4 ⊢ Rel (𝐴 ∩ (V × V))
16		dfrel2 5120	. . . 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 2219	. 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (𝐴 ∩ (V × V))
19	9, 10, 18	3eqtr2i 2223	1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))

Syntax hints:   = wceq 1364  Vcvv 2763   ∩ cin 3156   ⊆ wss 3157   × cxp 4661  ^◡ccnv 4662  Rel wrel 4668

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671

This theorem is referenced by:  cnvcnv2  5123  cnvcnvss  5124  structcnvcnv  12694  strslfv2d  12721