Theorem grpinvfval 9350

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 4207	. . . . 5 |- (G e. Grp -> ran G e. _V)
2		grpinvfval.1	. . . . 5 |- X = ran G
3	1, 2	syl5eqel 1975	. . . 4 |- (G e. Grp -> X e. _V)
4		opabex2g 4540	. . . 4 |- (X e. _V -> {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})} e. _V)
5	3, 4	syl 12	. . 3 |- (G e. Grp -> {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})} e. _V)
6		rneq 4186	. . . . . . . 8 |- (g = G -> ran g = ran G)
7	6, 2	syl6eqr 1946	. . . . . . 7 |- (g = G -> ran g = X)
8	7	eleq2d 1964	. . . . . 6 |- (g = G -> (x e. ran g <-> x e. X))
9		rabeq 2289	. . . . . . . . . 10 |- (ran g = X -> {y e. ran g | (ygx) = (Id` g)} = {y e. X | (ygx) = (Id` g)})
10	7, 9	syl 12	. . . . . . . . 9 |- (g = G -> {y e. ran g | (ygx) = (Id` g)} = {y e. X | (ygx) = (Id` g)})
11		opreq 4888	. . . . . . . . . . 11 |- (g = G -> (ygx) = (yGx))
12		fveq2 4681	. . . . . . . . . . . 12 |- (g = G -> (Id` g) = (Id` G))
13		grpinvfval.2	. . . . . . . . . . . 12 |- U = (Id` G)
14	12, 13	syl6eqr 1946	. . . . . . . . . . 11 |- (g = G -> (Id` g) = U)
15	11, 14	eqeq12d 1899	. . . . . . . . . 10 |- (g = G -> ((ygx) = (Id` g) <-> (yGx) = U))
16	15	rabbidv 2287	. . . . . . . . 9 |- (g = G -> {y e. X | (ygx) = (Id` g)} = {y e. X | (yGx) = U})
17	10, 16	eqtrd 1925	. . . . . . . 8 |- (g = G -> {y e. ran g | (ygx) = (Id` g)} = {y e. X | (yGx) = U})
18	17	unieqd 3188	. . . . . . 7 |- (g = G -> U.{y e. ran g | (ygx) = (Id` g)} = U.{y e. X | (yGx) = U})
19	18	eqeq2d 1895	. . . . . 6 |- (g = G -> (n = U.{y e. ran g | (ygx) = (Id` g)} <-> n = U.{y e. X | (yGx) = U}))
20	8, 19	anbi12d 690	. . . . 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 3401	. . . 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 9318	. . . 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 4742	. . 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 768	. 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 1940	1 |- (G e. Grp -> N = {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {crab 2108  _Vcvv 2292  U.cuni 3177  {copab 3395  ran crn 3987  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  invcgn 9313

This theorem is referenced by:  grpinvval 9351  grpinvf 9364  invtrgrp 14979

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-ginv 9318

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