Theorem opidon2 10371

Description: An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.)

Hypothesis

Ref	Expression
opidon2.1	|- X = ran G

Assertion

Ref	Expression
opidon2	|- (G e. (Magma i^i ExId ) -> G:(X X. X)-onto->X)

Proof of Theorem opidon2

Step	Hyp	Ref	Expression
1		eqid 1884	. . 3 |- dom dom G = dom dom G
2	1	opidon 10369	. 2 |- (G e. (Magma i^i ExId ) -> G:(dom dom G X. dom dom G)-onto->dom dom G)
3		forn 4620	. . . 4 |- (G:(dom dom G X. dom dom G)-onto->dom dom G -> ran G = dom dom G)
4		opidon2.1	. . . 4 |- X = ran G
5	3, 4	syl5req 1941	. . 3 |- (G:(dom dom G X. dom dom G)-onto->dom dom G -> dom dom G = X)
6		xpeq12 4020	. . . . . . 7 |- ((dom dom G = X /\ dom dom G = X) -> (dom dom G X. dom dom G) = (X X. X))
7	6	anidms 480	. . . . . 6 |- (dom dom G = X -> (dom dom G X. dom dom G) = (X X. X))
8		foeq2 4614	. . . . . 6 |- ((dom dom G X. dom dom G) = (X X. X) -> (G:(dom dom G X. dom dom G)-onto->dom dom G <-> G:(X X. X)-onto->dom dom G))
9	7, 8	syl 12	. . . . 5 |- (dom dom G = X -> (G:(dom dom G X. dom dom G)-onto->dom dom G <-> G:(X X. X)-onto->dom dom G))
10		foeq3 4615	. . . . 5 |- (dom dom G = X -> (G:(X X. X)-onto->dom dom G <-> G:(X X. X)-onto->X))
11	9, 10	bitrd 587	. . . 4 |- (dom dom G = X -> (G:(dom dom G X. dom dom G)-onto->dom dom G <-> G:(X X. X)-onto->X))
12	11	biimpd 170	. . 3 |- (dom dom G = X -> (G:(dom dom G X. dom dom G)-onto->dom dom G -> G:(X X. X)-onto->X))
13	5, 12	mpcom 60	. 2 |- (G:(dom dom G X. dom dom G)-onto->dom dom G -> G:(X X. X)-onto->X)
14	2, 13	syl 12	1 |- (G e. (Magma i^i ExId ) -> G:(X X. X)-onto->X)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300   i^i cin 2592   X. cxp 3984  dom cdm 3986  ran crn 3987  -onto->wfo 3996   ExId cexid 10361  Magmacmagm 10365

This theorem is referenced by:  exidreslem 16030

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-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-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-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-exid 10362  df-mgm 10366

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