Theorem grpinvval 8067

Description: The inverse of a group element.

Hypotheses

Ref	Expression
grpinvfval.1	|- X = ran G
grpinvfval.2	|- U = (Id` G)
grpinvfval.3	|- N = (inv` G)

Assertion

Ref	Expression
grpinvval	|- ((G e. Grp /\ A e. X) -> (N` A) = U.{y e. X | (yGA) = U})

Distinct variable groups:   y,A   y,G   y,X

Proof of Theorem grpinvval

Step	Hyp	Ref	Expression
1		grpinvfval.1	. . . . 5 |- X = ran G
2		grpinvfval.2	. . . . 5 |- U = (Id` G)
3		grpinvfval.3	. . . . 5 |- N = (inv` G)
4	1, 2, 3	grpinvfval 8066	. . . 4 |- (G e. Grp -> N = {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})})
5	4	fveq1d 3726	. . 3 |- (G e. Grp -> (N` A) = ({<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})}` A))
6	5	adantr 389	. 2 |- ((G e. Grp /\ A e. X) -> (N` A) = ({<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})}` A))
7		opreq2 3969	. . . . . . 7 |- (x = A -> (yGx) = (yGA))
8	7	eqeq1d 1483	. . . . . 6 |- (x = A -> ((yGx) = U <-> (yGA) = U))
9	8	rabbisdv 1807	. . . . 5 |- (x = A -> {y e. X | (yGx) = U} = {y e. X | (yGA) = U})
10	9	unieqd 2512	. . . 4 |- (x = A -> U.{y e. X | (yGx) = U} = U.{y e. X | (yGA) = U})
11		eqid 1475	. . . 4 |- {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})} = {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})}
12	10, 11	fvopab4g 3779	. . 3 |- ((A e. X /\ U.{y e. X | (yGA) = U} e. V) -> ({<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})}` A) = U.{y e. X | (yGA) = U})
13		pm3.27 323	. . 3 |- ((G e. Grp /\ A e. X) -> A e. X)
14		rnexg 3359	. . . . . 6 |- (G e. Grp -> ran G e. V)
15	14, 1	syl5eqel 1552	. . . . 5 |- (G e. Grp -> X e. V)
16		rabexg 2724	. . . . 5 |- (X e. V -> {y e. X | (yGA) = U} e. V)
17		uniexg 2871	. . . . 5 |- ({y e. X | (yGA) = U} e. V -> U.{y e. X | (yGA) = U} e. V)
18	15, 16, 17	3syl 20	. . . 4 |- (G e. Grp -> U.{y e. X | (yGA) = U} e. V)
19	18	adantr 389	. . 3 |- ((G e. Grp /\ A e. X) -> U.{y e. X | (yGA) = U} e. V)
20	12, 13, 19	sylanc 471	. 2 |- ((G e. Grp /\ A e. X) -> ({<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})}` A) = U.{y e. X | (yGA) = U})
21	6, 20	eqtrd 1507	1 |- ((G e. Grp /\ A e. X) -> (N` A) = U.{y e. X | (yGA) = U})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {crab 1648  Vcvv 1811  U.cuni 2503  {copab 2666  ran crn 3171  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Idcgi 8034  invcgn 8035

This theorem is referenced by:  grpinvcl 8068  grpinv 8069  addinv 8128

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-opr 3965  df-ginv 8039

Copyright terms: Public domain