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Theorem cnvcnv 4716

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 4646	. . . . 5 ⊢ Rel ^◡^◡A
2		df-rel 4295	. . . . 5 ⊢ (Rel ^◡^◡A ↔ ^◡^◡A ⊆ (V × V))
3	1, 2	mpbi 133	. . . 4 ⊢ ^◡^◡A ⊆ (V × V)
4		relxp 4390	. . . . 5 ⊢ Rel (V × V)
5		dfrel2 4714	. . . . 5 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
6	4, 5	mpbi 133	. . . 4 ⊢ ^◡^◡(V × V) = (V × V)
7	3, 6	sseqtr4i 2972	. . 3 ⊢ ^◡^◡A ⊆ ^◡^◡(V × V)
8		dfss 2926	. . 3 ⊢ (^◡^◡A ⊆ ^◡^◡(V × V) ↔ ^◡^◡A = (^◡^◡A ∩ ^◡^◡(V × V)))
9	7, 8	mpbi 133	. 2 ⊢ ^◡^◡A = (^◡^◡A ∩ ^◡^◡(V × V))
10		cnvin 4674	. 2 ⊢ ^◡(^◡A ∩ ^◡(V × V)) = (^◡^◡A ∩ ^◡^◡(V × V))
11		cnvin 4674	. . . 4 ⊢ ^◡(A ∩ (V × V)) = (^◡A ∩ ^◡(V × V))
12	11	cnveqi 4453	. . 3 ⊢ ^◡^◡(A ∩ (V × V)) = ^◡(^◡A ∩ ^◡(V × V))
13		inss2 3152	. . . . 5 ⊢ (A ∩ (V × V)) ⊆ (V × V)
14		df-rel 4295	. . . . 5 ⊢ (Rel (A ∩ (V × V)) ↔ (A ∩ (V × V)) ⊆ (V × V))
15	13, 14	mpbir 134	. . . 4 ⊢ Rel (A ∩ (V × V))
16		dfrel2 4714	. . . 4 ⊢ (Rel (A ∩ (V × V)) ↔ ^◡^◡(A ∩ (V × V)) = (A ∩ (V × V)))
17	15, 16	mpbi 133	. . 3 ⊢ ^◡^◡(A ∩ (V × V)) = (A ∩ (V × V))
18	12, 17	eqtr3i 2059	. 2 ⊢ ^◡(^◡A ∩ ^◡(V × V)) = (A ∩ (V × V))
19	9, 10, 18	3eqtr2i 2063	1 ⊢ ^◡^◡A = (A ∩ (V × V))

Colors of variables: wff set class

Syntax hints: = wceq 1242 Vcvv 2551 ∩ cin 2910 ⊆ wss 2911 × cxp 4286 ^◡ccnv 4287 Rel wrel 4293

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935

This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296

This theorem is referenced by: cnvcnv2 4717 cnvcnvss 4718