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Theorem cnvcnv 6212

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 6164	. . 3 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
2		cnvin 6164	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
3	2	cnveqi 5885	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
4		relcnv 6122	. . . . . 6 ⊢ Rel ^◡^◡𝐴
5		df-rel 5692	. . . . . 6 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
6	4, 5	mpbi 230	. . . . 5 ⊢ ^◡^◡𝐴 ⊆ (V × V)
7		relxp 5703	. . . . . 6 ⊢ Rel (V × V)
8		dfrel2 6209	. . . . . 6 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
9	7, 8	mpbi 230	. . . . 5 ⊢ ^◡^◡(V × V) = (V × V)
10	6, 9	sseqtrri 4033	. . . 4 ⊢ ^◡^◡𝐴 ⊆ ^◡^◡(V × V)
11		dfss 3970	. . . 4 ⊢ (^◡^◡𝐴 ⊆ ^◡^◡(V × V) ↔ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V)))
12	10, 11	mpbi 230	. . 3 ⊢ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
13	1, 3, 12	3eqtr4ri 2776	. 2 ⊢ ^◡^◡𝐴 = ^◡^◡(𝐴 ∩ (V × V))
14		relinxp 5824	. . 3 ⊢ Rel (𝐴 ∩ (V × V))
15		dfrel2 6209	. . 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 2765	1 ⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 × cxp 5683 ^◡ccnv 5684 Rel wrel 5690

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693

This theorem is referenced by: cnvcnv2 6213 cnvcnvss 6214 cnvrescnv 6215 structcnvcnv 17190 strfv2d 17238 elcnvcnvintab 43595 relintab 43596 nonrel 43597 elcnvcnvlem 43612 cnvcnvintabd 43613 tposresg 48778

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