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 6147

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 6099	. . 3 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
2		cnvin 6099	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
3	2	cnveqi 5819	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
4		relcnv 6063	. . . . . 6 ⊢ Rel ^◡^◡𝐴
5		df-rel 5628	. . . . . 6 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
6	4, 5	mpbi 232	. . . . 5 ⊢ ^◡^◡𝐴 ⊆ (V × V)
7		relxp 5639	. . . . . 6 ⊢ Rel (V × V)
8		dfrel2 6144	. . . . . 6 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
9	7, 8	mpbi 232	. . . . 5 ⊢ ^◡^◡(V × V) = (V × V)
10	6, 9	sseqtrri 3966	. . . 4 ⊢ ^◡^◡𝐴 ⊆ ^◡^◡(V × V)
11		dfss 3904	. . . 4 ⊢ (^◡^◡𝐴 ⊆ ^◡^◡(V × V) ↔ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V)))
12	10, 11	mpbi 232	. . 3 ⊢ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
13	1, 3, 12	3eqtr4ri 2775	. 2 ⊢ ^◡^◡𝐴 = ^◡^◡(𝐴 ∩ (V × V))
14		relinxp 5760	. . 3 ⊢ Rel (𝐴 ∩ (V × V))
15		dfrel2 6144	. . 3 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
16	14, 15	mpbi 232	. 2 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
17	13, 16	eqtri 2764	1 ⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

Colors of variables: wff setvar class

Syntax hints: = wceq 1548 Vcvv 3433 ∩ cin 3884 ⊆ wss 3885 × cxp 5619 ^◡ccnv 5620 Rel wrel 5626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-ext 2713 ax-sep 5221 ax-pr 5365

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-sb 2075 df-clab 2720 df-cleq 2733 df-clel 2816 df-rab 3394 df-v 3435 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-sn 4559 df-pr 4561 df-op 4565 df-br 5076 df-opab 5138 df-xp 5627 df-rel 5628 df-cnv 5629

This theorem is referenced by: cnvcnv2 6148 cnvcnvss 6149 cnvrescnv 6150 structcnvcnv 17118 strfv2d 17166 elcnvcnvintab 44041 relintab 44042 nonrel 44043 elcnvcnvlem 44058 cnvcnvintabd 44059 tposresg 49382

W3C validator