Theorem symgval 8670

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 8667	. . 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 3664	. 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 1113	. . . . . . . . 9 |- x = x
6	5	biantru 721	. . . . . . . 8 |- (x:A-1-1-onto->A <-> (x:A-1-1-onto->A /\ x = x))
7	6	abbii 1551	. . . . . . 7 |- {x | x:A-1-1-onto->A} = {x | (x:A-1-1-onto->A /\ x = x)}
8	4, 7	eqtr 1471	. . . . . 6 |- P = {x | (x:A-1-1-onto->A /\ x = x)}
9	8	f1oabexg 3639	. . . . 5 |- ((A e. V /\ A e. V) -> P e. V)
10	3, 3, 9	mp2an 694	. . . 4 |- P e. V
11	3, 4	symgoprab 8669	. . . 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 3960	. . 3 |- {<.<.f, g>., h>. | (f:A-1-1-onto->A /\ g:A-1-1-onto->A /\ h = (f o. g))} e. V
13		f1oeq2 3624	. . . . . 6 |- (x = A -> (f:x-1-1-onto->x <-> f:A-1-1-onto->x))
14		f1oeq3 3625	. . . . . 6 |- (x = A -> (f:A-1-1-onto->x <-> f:A-1-1-onto->A))
15	13, 14	bitrd 526	. . . . 5 |- (x = A -> (f:x-1-1-onto->x <-> f:A-1-1-onto->A))
16		f1oeq2 3624	. . . . . 6 |- (x = A -> (g:x-1-1-onto->x <-> g:A-1-1-onto->x))
17		f1oeq3 3625	. . . . . 6 |- (x = A -> (g:A-1-1-onto->x <-> g:A-1-1-onto->A))
18	16, 17	bitrd 526	. . . . 5 |- (x = A -> (g:x-1-1-onto->x <-> g:A-1-1-onto->A))
19	15, 18	3anbi12d 890	. . . 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 3935	. . 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 3729	. 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	3eqtr 1475	1 |- (SymGrp` A) = {<.<.f, g>., h>. | ((f e. P /\ g e. P) /\ h = (f o. g))}

Syntax hints:   /\ wa 223   /\ w3a 772   = wceq 1099   e. wcel 1105  {cab 1440  Vcvv 1786  {copab 2634   o. ccom 3137  -1-1-onto->wf1o 3144  ` cfv 3145  {copab2 3903  SymGrpcsymgrp 8666

This theorem is referenced by:  symgoprval 8671  symgf 8672

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-rex 1626  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-id 2797  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-f1 3158  df-fo 3159  df-f1o 3160  df-fv 3161  df-oprab 3905  df-symgrp 8667

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