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Theorem cnvcnv 5505

Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)

**Assertion**
Ref	Expression
cnvcnv	⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

**Proof of Theorem cnvcnv**
Step	Hyp	Ref	Expression
1		relcnv 5422	. . . . 5 ⊢ Rel ^◡^◡𝐴
2		df-rel 5045	. . . . 5 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
3	1, 2	mpbi 219	. . . 4 ⊢ ^◡^◡𝐴 ⊆ (V × V)
4		relxp 5150	. . . . 5 ⊢ Rel (V × V)
5		dfrel2 5502	. . . . 5 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
6	4, 5	mpbi 219	. . . 4 ⊢ ^◡^◡(V × V) = (V × V)
7	3, 6	sseqtr4i 3601	. . 3 ⊢ ^◡^◡𝐴 ⊆ ^◡^◡(V × V)
8		dfss 3555	. . 3 ⊢ (^◡^◡𝐴 ⊆ ^◡^◡(V × V) ↔ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V)))
9	7, 8	mpbi 219	. 2 ⊢ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
10		cnvin 5459	. 2 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
11		cnvin 5459	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
12	11	cnveqi 5219	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
13		inss2 3796	. . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V)
14		df-rel 5045	. . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
15	13, 14	mpbir 220	. . . 4 ⊢ Rel (𝐴 ∩ (V × V))
16		dfrel2 5502	. . . 4 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
17	15, 16	mpbi 219	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
18	12, 17	eqtr3i 2634	. 2 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (𝐴 ∩ (V × V))
19	9, 10, 18	3eqtr2i 2638	1 ⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

Colors of variables: wff setvar class

Syntax hints: = wceq 1475 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 × cxp 5036 ^◡ccnv 5037 Rel wrel 5043

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046

This theorem is referenced by: cnvcnv2 5506 cnvcnvss 5507 structcnvcnv 15706 strfv2d 15733 elcnvcnvintab 36907 relintab 36908 nonrel 36909 elcnvcnvlem 36924 cnvcnvintabd 36925