Theorem cnvcnv 5235

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 5160	. . . . 5 ⊢ Rel ◡◡𝐴
2		df-rel 4776	. . . . 5 ⊢ (Rel ◡◡𝐴 ↔ ◡◡𝐴 ⊆ (V × V))
3	1, 2	mpbi 145	. . . 4 ⊢ ◡◡𝐴 ⊆ (V × V)
4		relxp 4879	. . . . 5 ⊢ Rel (V × V)
5		dfrel2 5233	. . . . 5 ⊢ (Rel (V × V) ↔ ◡◡(V × V) = (V × V))
6	4, 5	mpbi 145	. . . 4 ⊢ ◡◡(V × V) = (V × V)
7	3, 6	sseqtrri 3283	. . 3 ⊢ ◡◡𝐴 ⊆ ◡◡(V × V)
8		dfss 3234	. . 3 ⊢ (◡◡𝐴 ⊆ ◡◡(V × V) ↔ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V)))
9	7, 8	mpbi 145	. 2 ⊢ ◡◡𝐴 = (◡◡𝐴 ∩ ◡◡(V × V))
10		cnvin 5190	. 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (◡◡𝐴 ∩ ◡◡(V × V))
11		cnvin 5190	. . . 4 ⊢ ◡(𝐴 ∩ (V × V)) = (◡𝐴 ∩ ◡(V × V))
12	11	cnveqi 4950	. . 3 ⊢ ◡◡(𝐴 ∩ (V × V)) = ◡(◡𝐴 ∩ ◡(V × V))
13		inss2 3452	. . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V)
14		df-rel 4776	. . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
15	13, 14	mpbir 146	. . . 4 ⊢ Rel (𝐴 ∩ (V × V))
16		dfrel2 5233	. . . 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 2261	. 2 ⊢ ◡(◡𝐴 ∩ ◡(V × V)) = (𝐴 ∩ (V × V))
19	9, 10, 18	3eqtr2i 2265	1 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))

Syntax hints:   = wceq 1402  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220   × cxp 4767  ^◡ccnv 4768  Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777

This theorem is referenced by:  cnvcnv2  5236  cnvcnvss  5237  structcnvcnv  13346  strslfv2d  13373