HomeHome Hilbert Space Explorer < Previous   Next >
Related theorems
Unicode version
Theorem symgval 10308
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
StepHypRef Expression
1 df-symgrp 10305 . . 3 |- SymGrp = {<.x, y>. | y = {<.<.f, g>., h>. | (f:x-1-1-onto->x /\ g:x-1-1-onto->x /\ h = (f o. g))}}
21fveq1i 3710 . 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 1122 . . . . . . . . 9 |- x = x
65biantru 722 . . . . . . . 8 |- (x:A-1-1-onto->A <-> (x:A-1-1-onto->A /\ x = x))
76abbii 1567 . . . . . . 7 |- {x | x:A-1-1-onto->A} = {x | (x:A-1-1-onto->A /\ x = x)}
84, 7eqtr 1487 . . . . . 6 |- P = {x | (x:A-1-1-onto->A /\ x = x)}
98f1oabexg 3685 . . . . 5 |- ((A e. V /\ A e. V) -> P e. V)
103, 3, 9mp2an 695 . . . 4 |- P e. V
113, 4symgoprab 10307 . . . 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))}
1210, 10, 11oprabex2 4006 . . 3 |- {<.<.f, g>., h>. | (f:A-1-1-onto->A /\ g:A-1-1-onto->A /\ h = (f o. g))} e. V
13 f1oeq2 3670 . . . . . 6 |- (x = A -> (f:x-1-1-onto->x <-> f:A-1-1-onto->x))
14 f1oeq3 3671 . . . . . 6 |- (x = A -> (f:A-1-1-onto->x <-> f:A-1-1-onto->A))
1513, 14bitrd 526 . . . . 5 |- (x = A -> (f:x-1-1-onto->x <-> f:A-1-1-onto->A))
16 f1oeq2 3670 . . . . . 6 |- (x = A -> (g:x-1-1-onto->x <-> g:A-1-1-onto->x))
17 f1oeq3 3671 . . . . . 6 |- (x = A -> (g:A-1-1-onto->x <-> g:A-1-1-onto->A))
1816, 17bitrd 526 . . . . 5 |- (x = A -> (g:x-1-1-onto->x <-> g:A-1-1-onto->A))
1915, 183anbi12d 891 . . . 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))))
2019oprabbidv 3981 . . 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))})
213, 12, 20fvopab 3775 . 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))}
222, 21, 113eqtr 1491 1 |- (SymGrp` A) = {<.<.f, g>., h>. | ((f e. P /\ g e. P) /\ h = (f o. g))}
Colors of variables: wff set class
Syntax hints:   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  {cab 1456  Vcvv 1802  {copab 2656   o. ccom 3164  -1-1-onto->wf1o 3171  ` cfv 3172  {copab2 3949  SymGrpcsymgrp 10304
This theorem is referenced by:  symgoprval 10309  symgf 10310
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-oprab 3951  df-symgrp 10305
Copyright terms: Public domain