Theorem mnlmxl2 14611

Description: The minimal elements of a preset are the maximal elements of the converse preset.

Assertion

Ref	Expression
mnlmxl2	|- (R e. Preset -> ( mnl ` R) = (mxl` `'R))

Proof of Theorem mnlmxl2

Step	Hyp	Ref	Expression
1		eqid 1884	. . . 4 |- dom R = dom R
2	1	domcnvpre 14574	. . 3 |- (R e. Preset -> dom R = dom `' R)
3	2	raleqdv 2269	. . . 4 |- (R e. Preset -> (A.y e. dom R(yRx -> y = x) <-> A.y e. dom `' R(yRx -> y = x)))
4		visset 2295	. . . . . . . 8 |- x e. _V
5		visset 2295	. . . . . . . 8 |- y e. _V
6		brcnvg 4142	. . . . . . . . 9 |- ((x e. _V /\ y e. _V) -> (x`'Ry <-> yRx))
7	6	bicomd 580	. . . . . . . 8 |- ((x e. _V /\ y e. _V) -> (yRx <-> x`'Ry))
8	4, 5, 7	mp2an 761	. . . . . . 7 |- (yRx <-> x`'Ry)
9		equcom 1488	. . . . . . 7 |- (y = x <-> x = y)
10	8, 9	imbi12i 205	. . . . . 6 |- ((yRx -> y = x) <-> (x`'Ry -> x = y))
11	10	a1i 8	. . . . 5 |- (R e. Preset -> ((yRx -> y = x) <-> (x`'Ry -> x = y)))
12	11	ralbidv 2123	. . . 4 |- (R e. Preset -> (A.y e. dom `' R(yRx -> y = x) <-> A.y e. dom `' R(x`'Ry -> x = y)))
13	3, 12	bitrd 587	. . 3 |- (R e. Preset -> (A.y e. dom R(yRx -> y = x) <-> A.y e. dom `' R(x`'Ry -> x = y)))
14	2, 13	rabeqbidv 2290	. 2 |- (R e. Preset -> {x e. dom R | A.y e. dom R(yRx -> y = x)} = {x e. dom `' R | A.y e. dom `' R(x`'Ry -> x = y)})
15	1	mnlelt2 14608	. 2 |- (R e. Preset -> ( mnl ` R) = {x e. dom R | A.y e. dom R(yRx -> y = x)})
16		dupre1 14584	. . 3 |- (R e. Preset -> `'R e. Preset )
17		eqid 1884	. . . 4 |- dom `' R = dom `' R
18	17	mxlelt2 14606	. . 3 |- (`'R e. Preset -> (mxl` `'R) = {x e. dom `' R | A.y e. dom `' R(x`'Ry -> x = y)})
19	16, 18	syl 12	. 2 |- (R e. Preset -> (mxl` `'R) = {x e. dom `' R | A.y e. dom `' R(x`'Ry -> x = y)})
20	14, 15, 19	3eqtr4d 1937	1 |- (R e. Preset -> ( mnl ` R) = (mxl` `'R))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108  _Vcvv 2292   class class class wbr 3338  `'ccnv 3985  dom cdm 3986  ` cfv 3998   Preset cpreset 14555  mxlcmxl 14556   mnl cmnl 14557

This theorem is referenced by:  mxlmnl2 14612

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-prs 14563  df-mxl 14589  df-mnl 14590

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