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Theorem cnvcnv 6180

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 6130	. . 3 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
2		cnvin 6130	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
3	2	cnveqi 5848	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
4		relcnv 6095	. . . . . 6 ⊢ Rel ^◡^◡𝐴
5		df-rel 5656	. . . . . 6 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
6	4, 5	mpbi 232	. . . . 5 ⊢ ^◡^◡𝐴 ⊆ (V × V)
7		relxp 5667	. . . . . 6 ⊢ Rel (V × V)
8		dfrel2 6177	. . . . . 6 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
9	7, 8	mpbi 232	. . . . 5 ⊢ ^◡^◡(V × V) = (V × V)
10	6, 9	sseqtrri 3987	. . . 4 ⊢ ^◡^◡𝐴 ⊆ ^◡^◡(V × V)
11		dfss 3925	. . . 4 ⊢ (^◡^◡𝐴 ⊆ ^◡^◡(V × V) ↔ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V)))
12	10, 11	mpbi 232	. . 3 ⊢ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
13	1, 3, 12	3eqtr4ri 2798	. 2 ⊢ ^◡^◡𝐴 = ^◡^◡(𝐴 ∩ (V × V))
14		relinxp 5789	. . 3 ⊢ Rel (𝐴 ∩ (V × V))
15		dfrel2 6177	. . 3 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
16	14, 15	mpbi 232	. 2 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
17	13, 16	eqtri 2787	1 ⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

Colors of variables: wff setvar class

Syntax hints: = wceq 1562 Vcvv 3456 ∩ cin 3905 ⊆ wss 3906 × cxp 5647 ^◡ccnv 5648 Rel wrel 5654

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-ext 2736 ax-sep 5248 ax-pr 5392

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-sb 2093 df-clab 2743 df-cleq 2756 df-clel 2839 df-rab 3417 df-v 3458 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-br 5103 df-opab 5165 df-xp 5655 df-rel 5656 df-cnv 5657

This theorem is referenced by: cnvcnv2 6181 cnvcnvssOLD 6183 cnvrescnv 6184 structcnvcnv 17191 strfv2d 17239 elcnvcnvintab 44163 relintab 44164 nonrel 44165 elcnvcnvlem 44180 cnvcnvintabd 44181 tposresg 49504

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