Theorem symgval 10202

Description: The value of the symmetry group function at A. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypotheses

Ref	Expression
elsymgrn.1	|- A e. _V
elsymgrn.2	|- P = {x | x:A-1-1-onto->A}

Assertion

Ref	Expression
symgval	|- (SymGrp` A) = {<.<.f, g>., h>. | ((f e. P /\ g e. P) /\ h = (f o. g))}

Distinct variable groups:   A,f,g,h,x   P,f,g,h

Proof of Theorem symgval

Step	Hyp	Ref	Expression
1		df-symgrp 10199	. . 3 |- SymGrp = {<.x, y>. | y = {<.<.f, g>., h>. | (f:x-1-1-onto->x /\ g:x-1-1-onto->x /\ h = (f o. g))}}
2	1	fveq1i 4682	. 2 |- (SymGrp` A) = ({<.x, y>. | y = {<.<.f, g>., h>. | (f:x-1-1-onto->x /\ g:x-1-1-onto->x /\ h = (f o. g))}}` A)
3		elsymgrn.1	. . 3 |- A e. _V
4		elsymgrn.2	. . . . . . 7 |- P = {x | x:A-1-1-onto->A}
5		equid 1484	. . . . . . . . 9 |- x = x
6	5	biantru 793	. . . . . . . 8 |- (x:A-1-1-onto->A <-> (x:A-1-1-onto->A /\ x = x))
7	6	abbii 2006	. . . . . . 7 |- {x | x:A-1-1-onto->A} = {x | (x:A-1-1-onto->A /\ x = x)}
8	4, 7	eqtri 1908	. . . . . 6 |- P = {x | (x:A-1-1-onto->A /\ x = x)}
9	8	f1oabexg 4650	. . . . 5 |- ((A e. _V /\ A e. _V) -> P e. _V)
10	3, 3, 9	mp2an 761	. . . 4 |- P e. _V
11	3, 4	symgoprab 10201	. . . 4 |- {<.<.f, g>., h>. | (f:A-1-1-onto->A /\ g:A-1-1-onto->A /\ h = (f o. g))} = {<.<.f, g>., h>. | ((f e. P /\ g e. P) /\ h = (f o. g))}
12	10, 10, 11	oprabex2 4950	. . 3 |- {<.<.f, g>., h>. | (f:A-1-1-onto->A /\ g:A-1-1-onto->A /\ h = (f o. g))} e. _V
13		f1oeq2 4631	. . . . . 6 |- (x = A -> (f:x-1-1-onto->x <-> f:A-1-1-onto->x))
14		f1oeq3 4632	. . . . . 6 |- (x = A -> (f:A-1-1-onto->x <-> f:A-1-1-onto->A))
15	13, 14	bitrd 587	. . . . 5 |- (x = A -> (f:x-1-1-onto->x <-> f:A-1-1-onto->A))
16		f1oeq2 4631	. . . . . 6 |- (x = A -> (g:x-1-1-onto->x <-> g:A-1-1-onto->x))
17		f1oeq3 4632	. . . . . 6 |- (x = A -> (g:A-1-1-onto->x <-> g:A-1-1-onto->A))
18	16, 17	bitrd 587	. . . . 5 |- (x = A -> (g:x-1-1-onto->x <-> g:A-1-1-onto->A))
19	15, 18	3anbi12d 1169	. . . 4 |- (x = A -> ((f:x-1-1-onto->x /\ g:x-1-1-onto->x /\ h = (f o. g)) <-> (f:A-1-1-onto->A /\ g:A-1-1-onto->A /\ h = (f o. g))))
20	19	oprabbidv 4922	. . 3 |- (x = A -> {<.<.f, g>., h>. | (f:x-1-1-onto->x /\ g:x-1-1-onto->x /\ h = (f o. g))} = {<.<.f, g>., h>. | (f:A-1-1-onto->A /\ g:A-1-1-onto->A /\ h = (f o. g))})
21	3, 12, 20	fvopab 4753	. 2 |- ({<.x, y>. | y = {<.<.f, g>., h>. | (f:x-1-1-onto->x /\ g:x-1-1-onto->x /\ h = (f o. g))}}` A) = {<.<.f, g>., h>. | (f:A-1-1-onto->A /\ g:A-1-1-onto->A /\ h = (f o. g))}
22	2, 21, 11	3eqtri 1912	1 |- (SymGrp` A) = {<.<.f, g>., h>. | ((f e. P /\ g e. P) /\ h = (f o. g))}

Syntax hints:   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292  {copab 3395   o. ccom 3990  -1-1-onto->wf1o 3997  ` cfv 3998  {copab2 4885  SymGrpcsymgrp 10198

This theorem is referenced by:  symgoprv 10203  symgf 10204

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-rep 3428  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-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-oprab 4887  df-symgrp 10199

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