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Theorem cnvcnv 5621

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 5575	. . 3 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
2		cnvin 5575	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
3	2	cnveqi 5329	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
4		relcnv 5538	. . . . . 6 ⊢ Rel ^◡^◡𝐴
5		df-rel 5150	. . . . . 6 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
6	4, 5	mpbi 220	. . . . 5 ⊢ ^◡^◡𝐴 ⊆ (V × V)
7		relxp 5160	. . . . . 6 ⊢ Rel (V × V)
8		dfrel2 5618	. . . . . 6 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
9	7, 8	mpbi 220	. . . . 5 ⊢ ^◡^◡(V × V) = (V × V)
10	6, 9	sseqtr4i 3671	. . . 4 ⊢ ^◡^◡𝐴 ⊆ ^◡^◡(V × V)
11		dfss 3622	. . . 4 ⊢ (^◡^◡𝐴 ⊆ ^◡^◡(V × V) ↔ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V)))
12	10, 11	mpbi 220	. . 3 ⊢ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
13	1, 3, 12	3eqtr4ri 2684	. 2 ⊢ ^◡^◡𝐴 = ^◡^◡(𝐴 ∩ (V × V))
14		inss2 3867	. . . 4 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V)
15		df-rel 5150	. . . 4 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
16	14, 15	mpbir 221	. . 3 ⊢ Rel (𝐴 ∩ (V × V))
17		dfrel2 5618	. . 3 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
18	16, 17	mpbi 220	. 2 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
19	13, 18	eqtri 2673	1 ⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

Colors of variables: wff setvar class

Syntax hints: = wceq 1523 Vcvv 3231 ∩ cin 3606 ⊆ wss 3607 × cxp 5141 ^◡ccnv 5142 Rel wrel 5148

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-xp 5149 df-rel 5150 df-cnv 5151

This theorem is referenced by: cnvcnv2 5623 cnvcnvss 5624 structcnvcnv 15918 strfv2d 15952 elcnvcnvintab 38205 relintab 38206 nonrel 38207 elcnvcnvlem 38222 cnvcnvintabd 38223

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