Theorem idtrgrp 14978

Description: The identity element of the trivial group.

Hypothesis

Ref	Expression
extopgrp.1	|- A e. B

Assertion

Ref	Expression
idtrgrp	|- (Id` {<.<.A, A>., A>.}) = A

Proof of Theorem idtrgrp

Step	Hyp	Ref	Expression
1		opex 3527	. . . . 5 |- <.A, A>. e. _V
2		extopgrp.1	. . . . . 6 |- A e. B
3	2	elisseti 2301	. . . . 5 |- A e. _V
4	1, 3	rnsnop 4375	. . . 4 |- ran {<.<.A, A>., A>.} = {A}
5	4	eqcomi 1888	. . 3 |- {A} = ran {<.<.A, A>., A>.}
6		eqid 1884	. . 3 |- (Id` {<.<.A, A>., A>.}) = (Id` {<.<.A, A>., A>.})
7	3	grpsn 9340	. . 3 |- {<.<.A, A>., A>.} e. Grp
8	5, 6, 7	grpidvallem 9341	. 2 |- (Id` {<.<.A, A>., A>.}) = U.{x e. {A} | A.y e. {A} (x{<.<.A, A>., A>.}y) = y}
9		elsni 3066	. . . . . . . 8 |- (x e. {A} -> x = A)
10	9	adantr 425	. . . . . . 7 |- ((x e. {A} /\ A.y e. {A} (x{<.<.A, A>., A>.}y) = y) -> x = A)
11		elsn 3058	. . . . . . . . 9 |- (x e. {A} <-> x = A)
12	11	biimpri 169	. . . . . . . 8 |- (x = A -> x e. {A})
13		elsni 3066	. . . . . . . . . . 11 |- (y e. {A} -> y = A)
14	1, 3	fopabsn 4815	. . . . . . . . . . . . . 14 |- {<.<.A, A>., A>.} = {<.x, y>. | (x e. {<.A, A>.} /\ y = A)}
15	14	opreqi 4896	. . . . . . . . . . . . 13 |- (A{<.<.A, A>., A>.}A) = (A{<.x, y>. | (x e. {<.A, A>.} /\ y = A)}A)
16		df-opr 4886	. . . . . . . . . . . . 13 |- (A{<.x, y>. | (x e. {<.A, A>.} /\ y = A)}A) = ({<.x, y>. | (x e. {<.A, A>.} /\ y = A)}` <.A, A>.)
17	1	snid 3069	. . . . . . . . . . . . . 14 |- <.A, A>. e. {<.A, A>.}
18		eqidd 1885	. . . . . . . . . . . . . . 15 |- (x = <.A, A>. -> A = A)
19		eqid 1884	. . . . . . . . . . . . . . 15 |- {<.x, y>. | (x e. {<.A, A>.} /\ y = A)} = {<.x, y>. | (x e. {<.A, A>.} /\ y = A)}
20	18, 19, 3	fvopab4 4743	. . . . . . . . . . . . . 14 |- (<.A, A>. e. {<.A, A>.} -> ({<.x, y>. | (x e. {<.A, A>.} /\ y = A)}` <.A, A>.) = A)
21	17, 20	ax-mp 7	. . . . . . . . . . . . 13 |- ({<.x, y>. | (x e. {<.A, A>.} /\ y = A)}` <.A, A>.) = A
22	15, 16, 21	3eqtri 1912	. . . . . . . . . . . 12 |- (A{<.<.A, A>., A>.}A) = A
23		opreq2 4890	. . . . . . . . . . . 12 |- (y = A -> (A{<.<.A, A>., A>.}y) = (A{<.<.A, A>., A>.}A))
24		id 73	. . . . . . . . . . . 12 |- (y = A -> y = A)
25	22, 23, 24	3eqtr4a 1954	. . . . . . . . . . 11 |- (y = A -> (A{<.<.A, A>., A>.}y) = y)
26	13, 25	syl 12	. . . . . . . . . 10 |- (y e. {A} -> (A{<.<.A, A>., A>.}y) = y)
27	26	rgen 2159	. . . . . . . . 9 |- A.y e. {A} (A{<.<.A, A>., A>.}y) = y
28		opreq1 4889	. . . . . . . . . . 11 |- (x = A -> (x{<.<.A, A>., A>.}y) = (A{<.<.A, A>., A>.}y))
29	28	eqeq1d 1892	. . . . . . . . . 10 |- (x = A -> ((x{<.<.A, A>., A>.}y) = y <-> (A{<.<.A, A>., A>.}y) = y))
30	29	ralbidv 2123	. . . . . . . . 9 |- (x = A -> (A.y e. {A} (x{<.<.A, A>., A>.}y) = y <-> A.y e. {A} (A{<.<.A, A>., A>.}y) = y))
31	27, 30	mpbiri 211	. . . . . . . 8 |- (x = A -> A.y e. {A} (x{<.<.A, A>., A>.}y) = y)
32	12, 31	jca 310	. . . . . . 7 |- (x = A -> (x e. {A} /\ A.y e. {A} (x{<.<.A, A>., A>.}y) = y))
33	10, 32	impbii 174	. . . . . 6 |- ((x e. {A} /\ A.y e. {A} (x{<.<.A, A>., A>.}y) = y) <-> x = A)
34	33	abbii 2006	. . . . 5 |- {x | (x e. {A} /\ A.y e. {A} (x{<.<.A, A>., A>.}y) = y)} = {x | x = A}
35		df-rab 2112	. . . . 5 |- {x e. {A} | A.y e. {A} (x{<.<.A, A>., A>.}y) = y} = {x | (x e. {A} /\ A.y e. {A} (x{<.<.A, A>., A>.}y) = y)}
36		df-sn 3049	. . . . 5 |- {A} = {x | x = A}
37	34, 35, 36	3eqtr4i 1921	. . . 4 |- {x e. {A} | A.y e. {A} (x{<.<.A, A>., A>.}y) = y} = {A}
38	37	unieqi 3187	. . 3 |- U.{x e. {A} | A.y e. {A} (x{<.<.A, A>., A>.}y) = y} = U.{A}
39	3	unisn 3193	. . 3 |- U.{A} = A
40	38, 39	eqtri 1908	. 2 |- U.{x e. {A} | A.y e. {A} (x{<.<.A, A>., A>.}y) = y} = A
41	8, 40	eqtri 1908	1 |- (Id` {<.<.A, A>., A>.}) = A

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  {crab 2108  {csn 3044  <.cop 3046  U.cuni 3177  {copab 3395  ran crn 3987  ` cfv 3998  (class class class)co 4884  Idcgi 9312

This theorem is referenced by:  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-3an 860  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-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-grp 9316  df-gid 9317

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