Theorem pi1bval 16088

Description: The base set of the fundamental group of a topological space at a given base point.

Hypothesis

Ref	Expression
pi1bval.1	|- X = U.J

Assertion

Ref	Expression
pi1bval	|- ((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))})

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

Proof of Theorem pi1bval

Step	Hyp	Ref	Expression
1		simpl 346	. . . 4 |- ((J e. Top /\ Y e. X) -> J e. Top)
2		simpr 350	. . . 4 |- ((J e. Top /\ Y e. X) -> Y e. X)
3		oprex 4907	. . . . . 6 |- (II Cn J) e. _V
4		df-sn 3049	. . . . . . . 8 |- {[f](~=ph` J)} = {g | g = [f](~=ph` J)}
5		snex 3492	. . . . . . . 8 |- {[f](~=ph` J)} e. _V
6	4, 5	eqeltrri 1968	. . . . . . 7 |- {g | g = [f](~=ph` J)} e. _V
7		simpr 350	. . . . . . . 8 |- ((((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J)) -> g = [f](~=ph` J))
8	7	ss2abi 2679	. . . . . . 7 |- {g | (((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))} C_ {g | g = [f](~=ph` J)}
9	6, 8	ssexi 3456	. . . . . 6 |- {g | (((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))} e. _V
10	3, 9	abrexex2 4847	. . . . 5 |- {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))} e. _V
11	10	a1i 8	. . . 4 |- ((J e. Top /\ Y e. X) -> {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))} e. _V)
12	1, 2, 11	3jca 1050	. . 3 |- ((J e. Top /\ Y e. X) -> (J e. Top /\ Y e. X /\ {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))} e. _V))
13		eleq1 1957	. . . . . 6 |- (j = J -> (j e. Top <-> J e. Top))
14		unieq 3185	. . . . . . 7 |- (j = J -> U.j = U.J)
15	14	eleq2d 1964	. . . . . 6 |- (j = J -> (y e. U.j <-> y e. U.J))
16	13, 15	anbi12d 690	. . . . 5 |- (j = J -> ((j e. Top /\ y e. U.j) <-> (J e. Top /\ y e. U.J)))
17		opreq2 4890	. . . . . . . 8 |- (j = J -> (II Cn j) = (II Cn J))
18		fveq2 4681	. . . . . . . . . . 11 |- (j = J -> (~=ph` j) = (~=ph` J))
19		eceq1 5335	. . . . . . . . . . 11 |- ((~=ph` j) = (~=ph` J) -> [f](~=ph` j) = [f](~=ph` J))
20	18, 19	syl 12	. . . . . . . . . 10 |- (j = J -> [f](~=ph` j) = [f](~=ph` J))
21	20	eqeq2d 1895	. . . . . . . . 9 |- (j = J -> (g = [f](~=ph` j) <-> g = [f](~=ph` J)))
22	21	anbi2d 678	. . . . . . . 8 |- (j = J -> ((((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` j)) <-> (((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` J))))
23	17, 22	rexeqbidv 2275	. . . . . . 7 |- (j = J -> (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))))
24	23	abbidv 2008	. . . . . 6 |- (j = J -> {g | E.f e. (II Cn j)(((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` j))} = {g | E.f e. (II Cn J)(((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` J))})
25	24	eqeq2d 1895	. . . . 5 |- (j = J -> (p = {g | E.f e. (II Cn j)(((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` j))} <-> p = {g | E.f e. (II Cn J)(((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` J))}))
26	16, 25	anbi12d 690	. . . 4 |- (j = J -> (((j e. Top /\ y e. U.j) /\ p = {g | E.f e. (II Cn j)(((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` j))}) <-> ((J e. Top /\ y e. U.J) /\ p = {g | E.f e. (II Cn J)(((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` J))})))
27		id 73	. . . . . . 7 |- (y = Y -> y = Y)
28		pi1bval.1	. . . . . . . . 9 |- X = U.J
29	28	eqcomi 1888	. . . . . . . 8 |- U.J = X
30	29	a1i 8	. . . . . . 7 |- (y = Y -> U.J = X)
31	27, 30	eleq12d 1965	. . . . . 6 |- (y = Y -> (y e. U.J <-> Y e. X))
32	31	anbi2d 678	. . . . 5 |- (y = Y -> ((J e. Top /\ y e. U.J) <-> (J e. Top /\ Y e. X)))
33		eqeq2 1893	. . . . . . . . . 10 |- (y = Y -> ((f` 0) = y <-> (f` 0) = Y))
34		eqeq2 1893	. . . . . . . . . 10 |- (y = Y -> ((f` 1) = y <-> (f` 1) = Y))
35	33, 34	anbi12d 690	. . . . . . . . 9 |- (y = Y -> (((f` 0) = y /\ (f` 1) = y) <-> ((f` 0) = Y /\ (f` 1) = Y)))
36	35	anbi1d 679	. . . . . . . 8 |- (y = Y -> ((((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` J)) <-> (((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))))
37	36	rexbidv 2124	. . . . . . 7 |- (y = Y -> (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))))
38	37	abbidv 2008	. . . . . 6 |- (y = Y -> {g | E.f e. (II Cn J)(((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` J))} = {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))})
39	38	eqeq2d 1895	. . . . 5 |- (y = Y -> (p = {g | E.f e. (II Cn J)(((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` J))} <-> p = {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))}))
40	32, 39	anbi12d 690	. . . 4 |- (y = Y -> (((J e. Top /\ y e. U.J) /\ p = {g | E.f e. (II Cn J)(((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` J))}) <-> ((J e. Top /\ Y e. X) /\ p = {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))})))
41		simpl 346	. . . . 5 |- (((J e. Top /\ Y e. X) /\ p = {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))}) -> (J e. Top /\ Y e. X))
42		pm3.21 306	. . . . 5 |- (p = {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))} -> ((J e. Top /\ Y e. X) -> ((J e. Top /\ Y e. X) /\ p = {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))})))
43	41, 42	impbid2 576	. . . 4 |- (p = {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))} -> (((J e. Top /\ Y e. X) /\ p = {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))}) <-> (J e. Top /\ Y e. X)))
44		moeq 2431	. . . . 5 |- E*p p = {g | E.f e. (II Cn j)(((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` j))}
45	44	moani 1820	. . . 4 |- E*p((j e. Top /\ y e. U.j) /\ p = {g | E.f e. (II Cn j)(((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` j))})
46		df-pi1b 16070	. . . 4 |- pi1b = {<.<.j, y>., p>. | ((j e. Top /\ y e. U.j) /\ p = {g | E.f e. (II Cn j)(((f` 0) = y /\ (f` 1) = y) /\ g = [f](~=ph` j))})}
47	26, 40, 43, 45, 46	oprabvaligg 10154	. . 3 |- ((J e. Top /\ Y e. X /\ {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))} e. _V) -> ((J e. Top /\ Y e. X) -> (Jpi1bY) = {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))}))
48	12, 47	mpcom 60	. 2 |- ((J e. Top /\ Y e. X) -> (Jpi1bY) = {g | E.f e. (II Cn J)(((f` 0) = Y /\ (f` 1) = Y) /\ g = [f](~=ph` J))})
49		df-opr 4886	. 2 |- (Jpi1bY) = (pi1b` <.J, Y>.)
50	48, 49	syl5eqr 1942	1 |- ((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))})

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  _Vcvv 2292  {csn 3044  <.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:  elpi1 16089

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

Copyright terms: Public domain