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Theorem cnvcnv 5490

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 5408	. . . . 5 ⊢ Rel ^◡^◡𝐴
2		df-rel 5034	. . . . 5 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
3	1, 2	mpbi 218	. . . 4 ⊢ ^◡^◡𝐴 ⊆ (V × V)
4		relxp 5138	. . . . 5 ⊢ Rel (V × V)
5		dfrel2 5487	. . . . 5 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
6	4, 5	mpbi 218	. . . 4 ⊢ ^◡^◡(V × V) = (V × V)
7	3, 6	sseqtr4i 3600	. . 3 ⊢ ^◡^◡𝐴 ⊆ ^◡^◡(V × V)
8		dfss 3554	. . 3 ⊢ (^◡^◡𝐴 ⊆ ^◡^◡(V × V) ↔ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V)))
9	7, 8	mpbi 218	. 2 ⊢ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
10		cnvin 5444	. 2 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
11		cnvin 5444	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
12	11	cnveqi 5206	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
13		inss2 3795	. . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V)
14		df-rel 5034	. . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
15	13, 14	mpbir 219	. . . 4 ⊢ Rel (𝐴 ∩ (V × V))
16		dfrel2 5487	. . . 4 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
17	15, 16	mpbi 218	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
18	12, 17	eqtr3i 2633	. 2 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (𝐴 ∩ (V × V))
19	9, 10, 18	3eqtr2i 2637	1 ⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

Colors of variables: wff setvar class

Syntax hints: = wceq 1474 Vcvv 3172 ∩ cin 3538 ⊆ wss 3539 × cxp 5025 ^◡ccnv 5026 Rel wrel 5032

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1712 ax-4 1727 ax-5 1826 ax-6 1874 ax-7 1921 ax-9 1985 ax-10 2005 ax-11 2020 ax-12 2033 ax-13 2233 ax-ext 2589 ax-sep 4703 ax-nul 4711 ax-pr 4827

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3an 1032 df-tru 1477 df-ex 1695 df-nf 1700 df-sb 1867 df-eu 2461 df-mo 2462 df-clab 2596 df-cleq 2602 df-clel 2605 df-nfc 2739 df-ral 2900 df-rex 2901 df-rab 2904 df-v 3174 df-dif 3542 df-un 3544 df-in 3546 df-ss 3553 df-nul 3874 df-if 4036 df-sn 4125 df-pr 4127 df-op 4131 df-br 4578 df-opab 4638 df-xp 5033 df-rel 5034 df-cnv 5035

This theorem is referenced by: cnvcnv2 5491 cnvcnvss 5492 structcnvcnv 15654 strfv2d 15681 elcnvcnvintab 36690 relintab 36691 nonrel 36692 elcnvcnvlem 36707 cnvcnvintabd 36708