cnvcnv - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > cnvcnv		GIF version

Theorem cnvcnv 5240

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 5165	. . . . 5 ⊢ Rel ^◡^◡𝐴
2		df-rel 4781	. . . . 5 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
3	1, 2	mpbi 145	. . . 4 ⊢ ^◡^◡𝐴 ⊆ (V × V)
4		relxp 4884	. . . . 5 ⊢ Rel (V × V)
5		dfrel2 5238	. . . . 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 5195	. 2 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
11		cnvin 5195	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
12	11	cnveqi 4955	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
13		inss2 3452	. . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V)
14		df-rel 4781	. . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
15	13, 14	mpbir 146	. . . 4 ⊢ Rel (𝐴 ∩ (V × V))
16		dfrel2 5238	. . . 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))

Colors of variables: wff set class

This proof depends on syntax axioms: = wceq 1402 Vcvv 2821 ∩ cin 3219 ⊆ wss 3220 × cxp 4772 ^◡ccnv 4773 Rel wrel 4779

This proof depends on 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 4249 ax-pow 4311 ax-pr 4346

This proof 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 3690 df-sn 3715 df-pr 3716 df-op 3718 df-br 4131 df-opab 4193 df-xp 4780 df-rel 4781 df-cnv 4782

This theorem is used by: cnvcnv2 5241 cnvcnvss 5242 structcnvcnv 13368 strslfv2d 13395