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Theorem cnvcnv 6192

Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.) (Proof shortened by BJ, 26-Nov-2021.)

**Assertion**
Ref	Expression
cnvcnv	⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

**Proof of Theorem cnvcnv**
Step	Hyp	Ref	Expression
1		cnvin 6144	. . 3 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
2		cnvin 6144	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
3	2	cnveqi 5865	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
4		relcnv 6102	. . . . . 6 ⊢ Rel ^◡^◡𝐴
5		df-rel 5672	. . . . . 6 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
6	4, 5	mpbi 230	. . . . 5 ⊢ ^◡^◡𝐴 ⊆ (V × V)
7		relxp 5683	. . . . . 6 ⊢ Rel (V × V)
8		dfrel2 6189	. . . . . 6 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
9	7, 8	mpbi 230	. . . . 5 ⊢ ^◡^◡(V × V) = (V × V)
10	6, 9	sseqtrri 4013	. . . 4 ⊢ ^◡^◡𝐴 ⊆ ^◡^◡(V × V)
11		dfss 3950	. . . 4 ⊢ (^◡^◡𝐴 ⊆ ^◡^◡(V × V) ↔ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V)))
12	10, 11	mpbi 230	. . 3 ⊢ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
13	1, 3, 12	3eqtr4ri 2768	. 2 ⊢ ^◡^◡𝐴 = ^◡^◡(𝐴 ∩ (V × V))
14		relinxp 5804	. . 3 ⊢ Rel (𝐴 ∩ (V × V))
15		dfrel2 6189	. . 3 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
16	14, 15	mpbi 230	. 2 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
17	13, 16	eqtri 2757	1 ⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 Vcvv 3463 ∩ cin 3930 ⊆ wss 3931 × cxp 5663 ^◡ccnv 5664 Rel wrel 5670

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pr 5412

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2713 df-cleq 2726 df-clel 2808 df-rab 3420 df-v 3465 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-sn 4607 df-pr 4609 df-op 4613 df-br 5124 df-opab 5186 df-xp 5671 df-rel 5672 df-cnv 5673

This theorem is referenced by: cnvcnv2 6193 cnvcnvss 6194 cnvrescnv 6195 structcnvcnv 17172 strfv2d 17220 elcnvcnvintab 43557 relintab 43558 nonrel 43559 elcnvcnvlem 43574 cnvcnvintabd 43575 tposresg 48737

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