Theorem opidon 10369

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

Hypothesis

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

Assertion

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

Proof of Theorem opidon

Step	Hyp	Ref	Expression
1		fooprv 4967	. 2 |- (G:(X X. X)-onto->X <-> (G:(X X. X)-->X /\ A.z e. X E.u e. X E.y e. X z = (uGy)))
2		inss1 2812	. . . 4 |- (Magma i^i ExId ) C_ Magma
3	2	sseli 2617	. . 3 |- (G e. (Magma i^i ExId ) -> G e. Magma)
4		opidon.1	. . . . 5 |- X = dom dom G
5	4	ismgm 10367	. . . 4 |- (G e. Magma -> (G e. Magma <-> G:(X X. X)-->X))
6	5	ibi 652	. . 3 |- (G e. Magma -> G:(X X. X)-->X)
7	3, 6	syl 12	. 2 |- (G e. (Magma i^i ExId ) -> G:(X X. X)-->X)
8		inss2 2813	. . . . 5 |- (Magma i^i ExId ) C_ ExId
9	8	sseli 2617	. . . 4 |- (G e. (Magma i^i ExId ) -> G e. ExId )
10	4	isexid 10364	. . . . 5 |- (G e. ExId -> (G e. ExId <-> E.u e. X A.x e. X ((xGu) = x /\ (uGx) = x)))
11	10	biimpd 170	. . . 4 |- (G e. ExId -> (G e. ExId -> E.u e. X A.x e. X ((xGu) = x /\ (uGx) = x)))
12	9, 9, 11	sylc 83	. . 3 |- (G e. (Magma i^i ExId ) -> E.u e. X A.x e. X ((xGu) = x /\ (uGx) = x))
13		opreq2 4890	. . . . . . . . . 10 |- (x = z -> (uGx) = (uGz))
14		id 73	. . . . . . . . . 10 |- (x = z -> x = z)
15	13, 14	eqeq12d 1899	. . . . . . . . 9 |- (x = z -> ((uGx) = x <-> (uGz) = z))
16	15	rcla4v 2376	. . . . . . . 8 |- (z e. X -> (A.x e. X (uGx) = x -> (uGz) = z))
17		opreq2 4890	. . . . . . . . . . . 12 |- (y = z -> (uGy) = (uGz))
18	17	eqeq1d 1892	. . . . . . . . . . 11 |- (y = z -> ((uGy) = z <-> (uGz) = z))
19		eqcom 1886	. . . . . . . . . . 11 |- (z = (uGy) <-> (uGy) = z)
20	18, 19	syl5bb 591	. . . . . . . . . 10 |- (y = z -> (z = (uGy) <-> (uGz) = z))
21	20	rcla4ev 2381	. . . . . . . . 9 |- ((z e. X /\ (uGz) = z) -> E.y e. X z = (uGy))
22	21	ex 402	. . . . . . . 8 |- (z e. X -> ((uGz) = z -> E.y e. X z = (uGy)))
23	16, 22	syld 30	. . . . . . 7 |- (z e. X -> (A.x e. X (uGx) = x -> E.y e. X z = (uGy)))
24		simpr 350	. . . . . . . 8 |- (((xGu) = x /\ (uGx) = x) -> (uGx) = x)
25	24	ralimi 2168	. . . . . . 7 |- (A.x e. X ((xGu) = x /\ (uGx) = x) -> A.x e. X (uGx) = x)
26	23, 25	syl5 20	. . . . . 6 |- (z e. X -> (A.x e. X ((xGu) = x /\ (uGx) = x) -> E.y e. X z = (uGy)))
27	26	reximdv 2202	. . . . 5 |- (z e. X -> (E.u e. X A.x e. X ((xGu) = x /\ (uGx) = x) -> E.u e. X E.y e. X z = (uGy)))
28	27	impcom 378	. . . 4 |- ((E.u e. X A.x e. X ((xGu) = x /\ (uGx) = x) /\ z e. X) -> E.u e. X E.y e. X z = (uGy))
29	28	r19.21aiva 2176	. . 3 |- (E.u e. X A.x e. X ((xGu) = x /\ (uGx) = x) -> A.z e. X E.u e. X E.y e. X z = (uGy))
30	12, 29	syl 12	. 2 |- (G e. (Magma i^i ExId ) -> A.z e. X E.u e. X E.y e. X z = (uGy))
31	1, 7, 30	sylanbrc 527	1 |- (G e. (Magma i^i ExId ) -> G:(X X. X)-onto->X)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   i^i cin 2592   X. cxp 3984  dom cdm 3986  -->wf 3994  -onto->wfo 3996  (class class class)co 4884   ExId cexid 10361  Magmacmagm 10365

This theorem is referenced by:  rngopid 10370  opidon2 10371  symgfo 14730  zintdom 14787

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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