Theorem symgval 10688

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 10685	. . 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 3836	. 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 1162	. . . . . . . . 9 |- x = x
6	5	biantru 729	. . . . . . . 8 |- (x:A-1-1-onto->A <-> (x:A-1-1-onto->A /\ x = x))
7	6	abbii 1618	. . . . . . 7 |- {x | x:A-1-1-onto->A} = {x | (x:A-1-1-onto->A /\ x = x)}
8	4, 7	eqtri 1538	. . . . . 6 |- P = {x | (x:A-1-1-onto->A /\ x = x)}
9	8	f1oabexg 3808	. . . . 5 |- ((A e. V /\ A e. V) -> P e. V)
10	3, 3, 9	mp2an 701	. . . 4 |- P e. V
11	3, 4	symgoprab 10687	. . . 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 4081	. . 3 |- {<.<.f, g>., h>. | (f:A-1-1-onto->A /\ g:A-1-1-onto->A /\ h = (f o. g))} e. V
13		f1oeq2 3793	. . . . . 6 |- (x = A -> (f:x-1-1-onto->x <-> f:A-1-1-onto->x))
14		f1oeq3 3794	. . . . . 6 |- (x = A -> (f:A-1-1-onto->x <-> f:A-1-1-onto->A))
15	13, 14	bitrd 531	. . . . 5 |- (x = A -> (f:x-1-1-onto->x <-> f:A-1-1-onto->A))
16		f1oeq2 3793	. . . . . 6 |- (x = A -> (g:x-1-1-onto->x <-> g:A-1-1-onto->x))
17		f1oeq3 3794	. . . . . 6 |- (x = A -> (g:A-1-1-onto->x <-> g:A-1-1-onto->A))
18	16, 17	bitrd 531	. . . . 5 |- (x = A -> (g:x-1-1-onto->x <-> g:A-1-1-onto->A))
19	15, 18	3anbi12d 900	. . . 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 4055	. . 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 3901	. 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 1542	1 |- (SymGrp` A) = {<.<.f, g>., h>. | ((f e. P /\ g e. P) /\ h = (f o. g))}

Syntax hints:   /\ wa 221   /\ w3a 781   = wceq 992   e. wcel 994  {cab 1505  Vcvv 1857  {copab 2740   o. ccom 3255  -1-1-onto->wf1o 3262  ` cfv 3263  {copab2 4022  SymGrpcsymgrp 10684

This theorem is referenced by:  symgoprv 10689  symgf 10690

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-rex 1696  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-oprab 4024  df-symgrp 10685

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