Theorem cnvcnv 5187

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 5112	. . . . 5 ⊢ Rel ◡◡𝐴
2		df-rel 4730	. . . . 5 ⊢ (Rel ◡◡𝐴 ↔ ◡◡𝐴 ⊆ (V × V))
3	1, 2	mpbi 145	. . . 4 ⊢ ◡◡𝐴 ⊆ (V × V)
4		relxp 4833	. . . . 5 ⊢ Rel (V × V)
5		dfrel2 5185	. . . . 5 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V))
6	4, 5	mpbi 145	. . . 4 ⊢ ◡◡(V × V) = (V × V)
7	3, 6	sseqtrri 3260	. . 3 ⊢ ◡◡𝐴 ⊆ ◡◡(V × V)
8		dfss 3212	. . 3 ⊢ (◡◡𝐴 ⊆ ◡◡(V × V) ↔ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V)))
9	7, 8	mpbi 145	. 2 ⊢ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V))
10		cnvin 5142	. 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (◡◡𝐴 ∩ ◡◡(V × V))
11		cnvin 5142	. . . 4 ⊢ ◡(𝐴 ∩ (V × V)) = (◡𝐴 ∩ ◡(V × V))
12	11	cnveqi 4903	. . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = ◡(◡𝐴 ∩ ◡(V × V))
13		inss2 3426	. . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V)
14		df-rel 4730	. . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
15	13, 14	mpbir 146	. . . 4 ⊢ Rel (𝐴 ∩ (V × V))
16		dfrel2 5185	. . . 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 2252	. 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (𝐴 ∩ (V × V))
19	9, 10, 18	3eqtr2i 2256	1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))

Syntax hints:   = wceq 1395  Vcvv 2800   ∩ cin 3197   ⊆ wss 3198   × cxp 4721  ^◡ccnv 4722  Rel wrel 4728

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731

This theorem is referenced by:  cnvcnv2  5188  cnvcnvss  5189  structcnvcnv  13088  strslfv2d  13115