Theorem grpinvfval 8062

Description: The inverse function of a group.

Hypotheses

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

Assertion

Ref	Expression
grpinvfval	|- (G e. Grp -> N = {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})})

Distinct variable groups:   x,n,y,G   n,X,x,y   U,n,x

Proof of Theorem grpinvfval

Step	Hyp	Ref	Expression
1		rnexg 3365	. . . . 5 |- (G e. Grp -> ran G e. V)
2		grpinvfval.1	. . . . 5 |- X = ran G
3	1, 2	syl5eqel 1555	. . . 4 |- (G e. Grp -> X e. V)
4		opabex2g 3617	. . . 4 |- (X e. V -> {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})} e. V)
5	3, 4	syl 10	. . 3 |- (G e. Grp -> {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})} e. V)
6		rneq 3345	. . . . . . . 8 |- (g = G -> ran g = ran G)
7	6, 2	syl6eqr 1528	. . . . . . 7 |- (g = G -> ran g = X)
8	7	eleq2d 1544	. . . . . 6 |- (g = G -> (x e. ran g <-> x e. X))
9		rabeq 1812	. . . . . . . . . 10 |- (ran g = X -> {y e. ran g | (ygx) = (Id` g)} = {y e. X | (ygx) = (Id` g)})
10	7, 9	syl 10	. . . . . . . . 9 |- (g = G -> {y e. ran g | (ygx) = (Id` g)} = {y e. X | (ygx) = (Id` g)})
11		opreq 3973	. . . . . . . . . . 11 |- (g = G -> (ygx) = (yGx))
12		fveq2 3730	. . . . . . . . . . . 12 |- (g = G -> (Id` g) = (Id` G))
13		grpinvfval.2	. . . . . . . . . . . 12 |- U = (Id` G)
14	12, 13	syl6eqr 1528	. . . . . . . . . . 11 |- (g = G -> (Id` g) = U)
15	11, 14	eqeq12d 1492	. . . . . . . . . 10 |- (g = G -> ((ygx) = (Id` g) <-> (yGx) = U))
16	15	rabbisdv 1810	. . . . . . . . 9 |- (g = G -> {y e. X | (ygx) = (Id` g)} = {y e. X | (yGx) = U})
17	10, 16	eqtrd 1510	. . . . . . . 8 |- (g = G -> {y e. ran g | (ygx) = (Id` g)} = {y e. X | (yGx) = U})
18	17	unieqd 2516	. . . . . . 7 |- (g = G -> U.{y e. ran g | (ygx) = (Id` g)} = U.{y e. X | (yGx) = U})
19	18	eqeq2d 1489	. . . . . 6 |- (g = G -> (n = U.{y e. ran g | (ygx) = (Id` g)} <-> n = U.{y e. X | (yGx) = U}))
20	8, 19	anbi12d 630	. . . . 5 |- (g = G -> ((x e. ran g /\ n = U.{y e. ran g | (ygx) = (Id` g)}) <-> (x e. X /\ n = U.{y e. X | (yGx) = U})))
21	20	opabbidv 2675	. . . 4 |- (g = G -> {<.x, n>. | (x e. ran g /\ n = U.{y e. ran g | (ygx) = (Id` g)})} = {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})})
22		df-ginv 8036	. . . 4 |- inv = {<.g, f>. | (g e. Grp /\ f = {<.x, n>. | (x e. ran g /\ n = U.{y e. ran g | (ygx) = (Id` g)})})}
23	21, 22	fvopab4g 3785	. . 3 |- ((G e. Grp /\ {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})} e. V) -> (inv` G) = {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})})
24	5, 23	mpdan 706	. 2 |- (G e. Grp -> (inv` G) = {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})})
25		grpinvfval.3	. 2 |- N = (inv` G)
26	24, 25	syl5eq 1522	1 |- (G e. Grp -> N = {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  {crab 1651  Vcvv 1814  U.cuni 2507  {copab 2671  ran crn 3177  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  Idcgi 8031  invcgn 8032

This theorem is referenced by:  grpinvval 8063  grpinvf 8075

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-opr 3971  df-ginv 8036

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