Theorem mxlmnl2 14612

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

Assertion

Ref	Expression
mxlmnl2	|- (R e. Preset -> (mxl` R) = ( mnl ` `'R))

Proof of Theorem mxlmnl2

Step	Hyp	Ref	Expression
1		preorel 14566	. . 3 |- (R e. Preset -> Rel R)
2		dfrel2 4358	. . . 4 |- (Rel R <-> `'`'R = R)
3	2	biimpi 168	. . 3 |- (Rel R -> `'`'R = R)
4		fveq2 4681	. . . . . 6 |- (R = `'`'R -> (mxl` R) = (mxl` `'`'R))
5	4	eqeq1d 1892	. . . . 5 |- (R = `'`'R -> ((mxl` R) = ( mnl ` `'R) <-> (mxl` `'`'R) = ( mnl ` `'R)))
6		dupre1 14584	. . . . . . 7 |- (R e. Preset -> `'R e. Preset )
7		mnlmxl2 14611	. . . . . . 7 |- (`'R e. Preset -> ( mnl ` `'R) = (mxl` `'`'R))
8	6, 7	syl 12	. . . . . 6 |- (R e. Preset -> ( mnl ` `'R) = (mxl` `'`'R))
9	8	eqcomd 1889	. . . . 5 |- (R e. Preset -> (mxl` `'`'R) = ( mnl ` `'R))
10	5, 9	syl5bir 227	. . . 4 |- (R = `'`'R -> (R e. Preset -> (mxl` R) = ( mnl ` `'R)))
11	10	eqcoms 1887	. . 3 |- (`'`'R = R -> (R e. Preset -> (mxl` R) = ( mnl ` `'R)))
12	1, 3, 11	3syl 24	. 2 |- (R e. Preset -> (R e. Preset -> (mxl` R) = ( mnl ` `'R)))
13	12	pm2.43i 78	1 |- (R e. Preset -> (mxl` R) = ( mnl ` `'R))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  `'ccnv 3985  Rel wrel 3991  ` cfv 3998   Preset cpreset 14555  mxlcmxl 14556   mnl cmnl 14557

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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