Theorem ismnd 10390

Description: The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.)

Hypothesis

Ref	Expression
ismond.1	|- X = dom dom G

Assertion

Ref	Expression
ismnd	|- (G e. A -> (G e. Mnd <-> (G e. SemiGrp /\ E.x e. X A.y e. X ((yGx) = y /\ (xGy) = y))))

Distinct variable group:   x,G,y

Proof of Theorem ismnd

Step	Hyp	Ref	Expression
1		ismond.1	. . . . 5 |- X = dom dom G
2	1	isexid 10364	. . . 4 |- (G e. A -> (G e. ExId <-> E.x e. X A.y e. X ((yGx) = y /\ (xGy) = y)))
3	2	anbi2d 678	. . 3 |- (G e. A -> ((G e. SemiGrp /\ G e. ExId ) <-> (G e. SemiGrp /\ E.x e. X A.y e. X ((yGx) = y /\ (xGy) = y))))
4		elin 2786	. . 3 |- (G e. (SemiGrp i^i ExId ) <-> (G e. SemiGrp /\ G e. ExId ))
5	3, 4	syl5bb 591	. 2 |- (G e. A -> (G e. (SemiGrp i^i ExId ) <-> (G e. SemiGrp /\ E.x e. X A.y e. X ((yGx) = y /\ (xGy) = y))))
6		df-mnd 10385	. . 3 |- Mnd = (SemiGrp i^i ExId )
7	6	eleq2i 1961	. 2 |- (G e. Mnd <-> G e. (SemiGrp i^i ExId ))
8	5, 7	syl5bb 591	1 |- (G e. A -> (G e. Mnd <-> (G e. SemiGrp /\ E.x e. X A.y e. X ((yGx) = y /\ (xGy) = y))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   i^i cin 2592  dom cdm 3986  (class class class)co 4884   ExId cexid 10361  SemiGrpcsem 10377  Mndcmnd 10384

This theorem is referenced by:  ismnd1 10391

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-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-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-exid 10362  df-mnd 10385

Copyright terms: Public domain