Theorem elpi1 16089

Description: The elements of the fundamental group.

Hypothesis

Ref	Expression
elpi1.1	|- X = U.J

Assertion

Ref	Expression
elpi1	|- ((J e. Top /\ Y e. X) -> (G e. (pi1b` <.J, Y>.) <-> E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ G = [f](~=ph` J))))

Distinct variable groups:   f,G   f,J   f,Y   f,X

Proof of Theorem elpi1

Step	Hyp	Ref	Expression
1		elpi1.1	. . . 4 |- X = U.J
2	1	pi1bval 16088	. . 3 |- ((J e. Top /\ Y e. X) -> (pi1b` <.J, Y>.) = {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))})
3	2	eleq2d 1964	. 2 |- ((J e. Top /\ Y e. X) -> (G e. (pi1b` <.J, Y>.) <-> G e. {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))}))
4		id 73	. . . . . 6 |- (G = [f](~=ph` J) -> G = [f](~=ph` J))
5		fvex 4689	. . . . . . 7 |- (~=ph` J) e. _V
6		ecexg 5322	. . . . . . 7 |- ((~=ph` J) e. _V -> [f](~=ph` J) e. _V)
7	5, 6	ax-mp 7	. . . . . 6 |- [f](~=ph` J) e. _V
8	4, 7	syl6eqel 1979	. . . . 5 |- (G = [f](~=ph` J) -> G e. _V)
9	8	ad2antll 443	. . . 4 |- ((f e. (II Cn J) /\ (((f` 0) = Y /\ (f` 1) = Y) /\ G = [f](~=ph` J))) -> G e. _V)
10	9	r19.23aiva 2212	. . 3 |- (E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ G = [f](~=ph` J)) -> G e. _V)
11		eqeq1 1890	. . . . 5 |- (g = G -> (g = [f](~=ph` J) <-> G = [f](~=ph` J)))
12	11	anbi2d 678	. . . 4 |- (g = G -> ((((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J)) <-> (((f` 0) = Y /\ (f` 1) = Y) /\ G = [f](~=ph` J))))
13	12	rexbidv 2124	. . 3 |- (g = G -> (E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J)) <-> E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ G = [f](~=ph` J))))
14	10, 13	elab3 2412	. 2 |- (G e. {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))} <-> E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ G = [f](~=ph` J)))
15	3, 14	syl6bb 595	1 |- ((J e. Top /\ Y e. X) -> (G e. (pi1b` <.J, Y>.) <-> E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ G = [f](~=ph` J))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  _Vcvv 2292  <.cop 3046  U.cuni 3177  ` cfv 3998  (class class class)co 4884  [cec 5316  0cc0 6386  1c1 6387  Topctop 8857   Cn ccn 9028  IIcii 15865  ~=phcphtpc 16044  pi1bcpi1b 16066

This theorem is referenced by:  elpi1i 16090  pi1bvalqs 16091  pi1gp 16095

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-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-iun 3257  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-fv 4014  df-opr 4886  df-oprab 4887  df-ec 5320  df-pi1b 16070

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