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Theorem cnvcnv 6189

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 6139	. . 3 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
2		cnvin 6139	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
3	2	cnveqi 5858	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
4		relcnv 6104	. . . . . 6 ⊢ Rel ^◡^◡𝐴
5		df-rel 5666	. . . . . 6 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
6	4, 5	mpbi 233	. . . . 5 ⊢ ^◡^◡𝐴 ⊆ (V × V)
7		relxp 5677	. . . . . 6 ⊢ Rel (V × V)
8		dfrel2 6186	. . . . . 6 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
9	7, 8	mpbi 233	. . . . 5 ⊢ ^◡^◡(V × V) = (V × V)
10	6, 9	sseqtrri 3983	. . . 4 ⊢ ^◡^◡𝐴 ⊆ ^◡^◡(V × V)
11		dfss 3921	. . . 4 ⊢ (^◡^◡𝐴 ⊆ ^◡^◡(V × V) ↔ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V)))
12	10, 11	mpbi 233	. . 3 ⊢ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
13	1, 3, 12	3eqtr4ri 2796	. 2 ⊢ ^◡^◡𝐴 = ^◡^◡(𝐴 ∩ (V × V))
14		relinxp 5799	. . 3 ⊢ Rel (𝐴 ∩ (V × V))
15		dfrel2 6186	. . 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 2785	1 ⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 Vcvv 3453 ∩ cin 3901 ⊆ wss 3902 × cxp 5657 ^◡ccnv 5658 Rel wrel 5664

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-ext 2734 ax-sep 5255 ax-pr 5402

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2741 df-cleq 2754 df-clel 2837 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-br 5108 df-opab 5172 df-xp 5665 df-rel 5666 df-cnv 5667

This theorem is used by: cnvcnv2 6190 cnvcnvssOLD 6192 cnvrescnv 6193 structcnvcnv 17249 strfv2d 17297 elcnvcnvintab 44409 relintab 44410 nonrel 44411 elcnvcnvlem 44426 cnvcnvintabd 44427 tposresg 49791

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