cnvcnv - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cnvcnv		Structured version Visualization version GIF version

Theorem cnvcnv 5803

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 5757	. . 3 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
2		cnvin 5757	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
3	2	cnveqi 5500	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
4		relcnv 5720	. . . . . 6 ⊢ Rel ^◡^◡𝐴
5		df-rel 5319	. . . . . 6 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
6	4, 5	mpbi 222	. . . . 5 ⊢ ^◡^◡𝐴 ⊆ (V × V)
7		relxp 5330	. . . . . 6 ⊢ Rel (V × V)
8		dfrel2 5800	. . . . . 6 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
9	7, 8	mpbi 222	. . . . 5 ⊢ ^◡^◡(V × V) = (V × V)
10	6, 9	sseqtr4i 3834	. . . 4 ⊢ ^◡^◡𝐴 ⊆ ^◡^◡(V × V)
11		dfss 3784	. . . 4 ⊢ (^◡^◡𝐴 ⊆ ^◡^◡(V × V) ↔ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V)))
12	10, 11	mpbi 222	. . 3 ⊢ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
13	1, 3, 12	3eqtr4ri 2832	. 2 ⊢ ^◡^◡𝐴 = ^◡^◡(𝐴 ∩ (V × V))
14		inss2 4029	. . . 4 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V)
15		df-rel 5319	. . . 4 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
16	14, 15	mpbir 223	. . 3 ⊢ Rel (𝐴 ∩ (V × V))
17		dfrel2 5800	. . 3 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
18	16, 17	mpbi 222	. 2 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
19	13, 18	eqtri 2821	1 ⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

Colors of variables: wff setvar class

Syntax hints: = wceq 1653 Vcvv 3385 ∩ cin 3768 ⊆ wss 3769 × cxp 5310 ^◡ccnv 5311 Rel wrel 5317

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-rel 5319 df-cnv 5320

This theorem is referenced by: cnvcnv2 5804 cnvcnvss 5805 structcnvcnv 16198 strfv2d 16230 elcnvcnvintab 38671 relintab 38672 nonrel 38673 elcnvcnvlem 38688 cnvcnvintabd 38689

W3C validator