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Theorem cnvcnv 6183

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 6134	. . 3 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
2		cnvin 6134	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
3	2	cnveqi 5851	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
4		relcnv 6099	. . . . . 6 ⊢ Rel ^◡^◡𝐴
5		df-rel 5657	. . . . . 6 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
6	4, 5	mpbi 233	. . . . 5 ⊢ ^◡^◡𝐴 ⊆ (V × V)
7		relxp 5668	. . . . . 6 ⊢ Rel (V × V)
8		dfrel2 6180	. . . . . 6 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
9	7, 8	mpbi 233	. . . . 5 ⊢ ^◡^◡(V × V) = (V × V)
10	6, 9	sseqtrri 3957	. . . 4 ⊢ ^◡^◡𝐴 ⊆ ^◡^◡(V × V)
11		dfss 4095	. . . 4 ⊢ (^◡^◡𝐴 ⊆ ^◡^◡(V × V) ↔ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V)))
12	10, 11	mpbi 233	. . 3 ⊢ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
13	1, 3, 12	3eqtr4ri 2794	. 2 ⊢ ^◡^◡𝐴 = ^◡^◡(𝐴 ∩ (V × V))
14		relinxp 5791	. . 3 ⊢ Rel (𝐴 ∩ (V × V))
15		dfrel2 6180	. . 3 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
16	14, 15	mpbi 233	. 2 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
17	13, 16	eqtri 2783	1 ⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1569 Vcvv 3450 ⊆ wss 3895 ∩ cin 4089 × cxp 5648 ^◡ccnv 5649 Rel wrel 5655

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-ext 2732 ax-sep 5248 ax-pr 5390

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-sb 2099 df-clab 2739 df-cleq 2752 df-clel 2835 df-rab 3413 df-v 3452 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4584 df-pr 4586 df-op 4590 df-br 5103 df-opab 5167 df-xp 5656 df-rel 5657 df-cnv 5658

This theorem is used by: cnvcnv2 6184 cnvcnvssOLD 6186 cnvrescnv 6187 structcnvcnv 17331 strfv2d 17379 elcnvcnvintab 44583 relintab 44584 nonrel 44585 elcnvcnvlem 44599 cnvcnvintabd 44600 tposresg 49986

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