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Theorem sconpi1 27128
Description: A path-connected topological space is simply connected iff its fundamental group is trivial. (Contributed by Mario Carneiro, 12-Feb-2015.)
Hypothesis
Ref Expression
sconpi1.1  |-  X  = 
U. J
Assertion
Ref Expression
sconpi1  |-  ( ( J  e. PCon  /\  Y  e.  X )  ->  ( J  e. SCon  <->  ( Base `  ( J  pi1  Y ) )  ~~  1o ) )

Proof of Theorem sconpi1
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scontop 27117 . . . . . . . . 9  |-  ( J  e. SCon  ->  J  e.  Top )
21adantl 466 . . . . . . . 8  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  J  e.  Top )
3 simpl 457 . . . . . . . 8  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  Y  e.  X )
4 eqid 2443 . . . . . . . . 9  |-  ( J  pi1  Y )  =  ( J  pi1  Y )
5 eqid 2443 . . . . . . . . 9  |-  ( Base `  ( J  pi1  Y ) )  =  ( Base `  ( J  pi1  Y ) )
6 simpl 457 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  Y  e.  X )  ->  J  e.  Top )
7 sconpi1.1 . . . . . . . . . . 11  |-  X  = 
U. J
87toptopon 18538 . . . . . . . . . 10  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
96, 8sylib 196 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  Y  e.  X )  ->  J  e.  (TopOn `  X ) )
10 simpr 461 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  Y  e.  X )  ->  Y  e.  X )
114, 5, 9, 10elpi1 20617 . . . . . . . 8  |-  ( ( J  e.  Top  /\  Y  e.  X )  ->  ( x  e.  (
Base `  ( J  pi1  Y )
)  <->  E. f  e.  ( II  Cn  J ) ( ( ( f `
 0 )  =  Y  /\  ( f `
 1 )  =  Y )  /\  x  =  [ f ] ( 
~=ph  `  J ) ) ) )
122, 3, 11syl2anc 661 . . . . . . 7  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  ( x  e.  ( Base `  ( J  pi1  Y ) )  <->  E. f  e.  ( II  Cn  J
) ( ( ( f `  0 )  =  Y  /\  (
f `  1 )  =  Y )  /\  x  =  [ f ] ( 
~=ph  `  J ) ) ) )
13 phtpcer 20567 . . . . . . . . . . . . 13  |-  (  ~=ph  `  J )  Er  (
II  Cn  J )
1413a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  (  ~=ph  `  J )  Er  (
II  Cn  J )
)
15 simpllr 758 . . . . . . . . . . . . . 14  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  J  e. SCon )
16 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  f  e.  ( II  Cn  J
) )
17 simprl 755 . . . . . . . . . . . . . . 15  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  ( f `  0 )  =  Y )
18 simprr 756 . . . . . . . . . . . . . . 15  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  ( f `  1 )  =  Y )
1917, 18eqtr4d 2478 . . . . . . . . . . . . . 14  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  ( f `  0 )  =  ( f `  1
) )
20 sconpht 27118 . . . . . . . . . . . . . 14  |-  ( ( J  e. SCon  /\  f  e.  ( II  Cn  J
)  /\  ( f `  0 )  =  ( f `  1
) )  ->  f
(  ~=ph  `  J )
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) )
2115, 16, 19, 20syl3anc 1218 . . . . . . . . . . . . 13  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  f (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) )
2217sneqd 3889 . . . . . . . . . . . . . 14  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  { (
f `  0 ) }  =  { Y } )
2322xpeq2d 4864 . . . . . . . . . . . . 13  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  ( (
0 [,] 1 )  X.  { ( f `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { Y }
) )
2421, 23breqtrd 4316 . . . . . . . . . . . 12  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  f (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { Y }
) )
2514, 24erthi 7147 . . . . . . . . . . 11  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  [ f ] (  ~=ph  `  J
)  =  [ ( ( 0 [,] 1
)  X.  { Y } ) ] ( 
~=ph  `  J ) )
262, 8sylib 196 . . . . . . . . . . . . 13  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  J  e.  (TopOn `  X
) )
27 eqid 2443 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 )  X.  { Y }
)  =  ( ( 0 [,] 1 )  X.  { Y }
)
284, 27pi1id 20623 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  X )  /\  Y  e.  X )  ->  [ ( ( 0 [,] 1
)  X.  { Y } ) ] ( 
~=ph  `  J )  =  ( 0g `  ( J  pi1  Y ) ) )
2926, 3, 28syl2anc 661 . . . . . . . . . . . 12  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  [ ( ( 0 [,] 1 )  X.  { Y } ) ] ( 
~=ph  `  J )  =  ( 0g `  ( J  pi1  Y ) ) )
3029ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  [ (
( 0 [,] 1
)  X.  { Y } ) ] ( 
~=ph  `  J )  =  ( 0g `  ( J  pi1  Y ) ) )
3125, 30eqtrd 2475 . . . . . . . . . 10  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  [ f ] (  ~=ph  `  J
)  =  ( 0g
`  ( J  pi1  Y ) ) )
32 elsn 3891 . . . . . . . . . . 11  |-  ( x  e.  { ( 0g
`  ( J  pi1  Y ) ) }  <->  x  =  ( 0g `  ( J  pi1  Y ) ) )
33 eqeq1 2449 . . . . . . . . . . 11  |-  ( x  =  [ f ] (  ~=ph  `  J )  ->  ( x  =  ( 0g `  ( J  pi1  Y ) )  <->  [ f ] ( 
~=ph  `  J )  =  ( 0g `  ( J  pi1  Y ) ) ) )
3432, 33syl5bb 257 . . . . . . . . . 10  |-  ( x  =  [ f ] (  ~=ph  `  J )  ->  ( x  e. 
{ ( 0g `  ( J  pi1  Y ) ) }  <->  [ f ] ( 
~=ph  `  J )  =  ( 0g `  ( J  pi1  Y ) ) ) )
3531, 34syl5ibrcom 222 . . . . . . . . 9  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  ( x  =  [ f ] ( 
~=ph  `  J )  ->  x  e.  { ( 0g `  ( J  pi1  Y ) ) } ) )
3635expimpd 603 . . . . . . . 8  |-  ( ( ( Y  e.  X  /\  J  e. SCon )  /\  f  e.  ( II  Cn  J ) )  -> 
( ( ( ( f `  0 )  =  Y  /\  (
f `  1 )  =  Y )  /\  x  =  [ f ] ( 
~=ph  `  J ) )  ->  x  e.  {
( 0g `  ( J  pi1  Y ) ) } ) )
3736rexlimdva 2841 . . . . . . 7  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  ( E. f  e.  ( II  Cn  J ) ( ( ( f `
 0 )  =  Y  /\  ( f `
 1 )  =  Y )  /\  x  =  [ f ] ( 
~=ph  `  J ) )  ->  x  e.  {
( 0g `  ( J  pi1  Y ) ) } ) )
3812, 37sylbid 215 . . . . . 6  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  ( x  e.  ( Base `  ( J  pi1  Y ) )  ->  x  e.  { ( 0g `  ( J  pi1  Y ) ) } ) )
3938ssrdv 3362 . . . . 5  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  (
Base `  ( J  pi1  Y )
)  C_  { ( 0g `  ( J  pi1  Y ) ) } )
404pi1grp 20622 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  Y  e.  X )  ->  ( J  pi1  Y )  e.  Grp )
4126, 3, 40syl2anc 661 . . . . . . 7  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  ( J  pi1  Y )  e.  Grp )
42 eqid 2443 . . . . . . . 8  |-  ( 0g
`  ( J  pi1  Y ) )  =  ( 0g `  ( J  pi1  Y ) )
435, 42grpidcl 15566 . . . . . . 7  |-  ( ( J  pi1  Y )  e.  Grp  ->  ( 0g `  ( J  pi1  Y ) )  e.  ( Base `  ( J  pi1  Y ) ) )
4441, 43syl 16 . . . . . 6  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  ( 0g `  ( J  pi1  Y ) )  e.  ( Base `  ( J  pi1  Y ) ) )
4544snssd 4018 . . . . 5  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  { ( 0g `  ( J  pi1  Y ) ) }  C_  ( Base `  ( J  pi1  Y ) ) )
4639, 45eqssd 3373 . . . 4  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  (
Base `  ( J  pi1  Y )
)  =  { ( 0g `  ( J  pi1  Y ) ) } )
47 fvex 5701 . . . . 5  |-  ( 0g
`  ( J  pi1  Y ) )  e.  _V
4847ensn1 7373 . . . 4  |-  { ( 0g `  ( J  pi1  Y ) ) }  ~~  1o
4946, 48syl6eqbr 4329 . . 3  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  (
Base `  ( J  pi1  Y )
)  ~~  1o )
5049adantll 713 . 2  |-  ( ( ( J  e. PCon  /\  Y  e.  X )  /\  J  e. SCon )  -> 
( Base `  ( J  pi1  Y )
)  ~~  1o )
51 simpll 753 . . 3  |-  ( ( ( J  e. PCon  /\  Y  e.  X )  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  ->  J  e. PCon )
52 eqid 2443 . . . . . . . . 9  |-  ( J  pi1  ( f `
 0 ) )  =  ( J  pi1  ( f ` 
0 ) )
53 eqid 2443 . . . . . . . . 9  |-  ( Base `  ( J  pi1 
( f `  0
) ) )  =  ( Base `  ( J  pi1  ( f `
 0 ) ) )
54 simplll 757 . . . . . . . . . . 11  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  J  e. PCon )
55 pcontop 27114 . . . . . . . . . . 11  |-  ( J  e. PCon  ->  J  e.  Top )
5654, 55syl 16 . . . . . . . . . 10  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  J  e.  Top )
5756, 8sylib 196 . . . . . . . . 9  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  J  e.  (TopOn `  X ) )
58 simprl 755 . . . . . . . . . . 11  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  f  e.  ( II  Cn  J
) )
59 iiuni 20457 . . . . . . . . . . . 12  |-  ( 0 [,] 1 )  = 
U. II
6059, 7cnf 18850 . . . . . . . . . . 11  |-  ( f  e.  ( II  Cn  J )  ->  f : ( 0 [,] 1 ) --> X )
6158, 60syl 16 . . . . . . . . . 10  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  f :
( 0 [,] 1
) --> X )
62 0elunit 11403 . . . . . . . . . 10  |-  0  e.  ( 0 [,] 1
)
63 ffvelrn 5841 . . . . . . . . . 10  |-  ( ( f : ( 0 [,] 1 ) --> X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( f `  0 )  e.  X )
6461, 62, 63sylancl 662 . . . . . . . . 9  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( f `  0 )  e.  X )
65 eqidd 2444 . . . . . . . . 9  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( f `  0 )  =  ( f `  0
) )
66 simprr 756 . . . . . . . . . 10  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( f `  0 )  =  ( f `  1
) )
6766eqcomd 2448 . . . . . . . . 9  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( f `  1 )  =  ( f `  0
) )
6852, 53, 57, 64, 58, 65, 67elpi1i 20618 . . . . . . . 8  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  [ f ] (  ~=ph  `  J
)  e.  ( Base `  ( J  pi1 
( f `  0
) ) ) )
69 eqid 2443 . . . . . . . . . . . . 13  |-  ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } )
7069pcoptcl 20593 . . . . . . . . . . . 12  |-  ( ( J  e.  (TopOn `  X )  /\  (
f `  0 )  e.  X )  ->  (
( ( 0 [,] 1 )  X.  {
( f `  0
) } )  e.  ( II  Cn  J
)  /\  ( (
( 0 [,] 1
)  X.  { ( f `  0 ) } ) `  0
)  =  ( f `
 0 )  /\  ( ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) `
 1 )  =  ( f `  0
) ) )
7157, 64, 70syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( (
( 0 [,] 1
)  X.  { ( f `  0 ) } )  e.  ( II  Cn  J )  /\  ( ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) `  0 )  =  ( f ` 
0 )  /\  (
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) ` 
1 )  =  ( f `  0 ) ) )
7271simp1d 1000 . . . . . . . . . 10  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( (
0 [,] 1 )  X.  { ( f `
 0 ) } )  e.  ( II 
Cn  J ) )
7371simp2d 1001 . . . . . . . . . 10  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( (
( 0 [,] 1
)  X.  { ( f `  0 ) } ) `  0
)  =  ( f `
 0 ) )
7471simp3d 1002 . . . . . . . . . 10  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( (
( 0 [,] 1
)  X.  { ( f `  0 ) } ) `  1
)  =  ( f `
 0 ) )
7552, 53, 57, 64, 72, 73, 74elpi1i 20618 . . . . . . . . 9  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  [ (
( 0 [,] 1
)  X.  { ( f `  0 ) } ) ] ( 
~=ph  `  J )  e.  ( Base `  ( J  pi1  ( f `
 0 ) ) ) )
76 simpllr 758 . . . . . . . . . . . 12  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  Y  e.  X )
777, 52, 4, 53, 5pconpi1 27126 . . . . . . . . . . . 12  |-  ( ( J  e. PCon  /\  (
f `  0 )  e.  X  /\  Y  e.  X )  ->  ( J  pi1  ( f `
 0 ) ) 
~=ph𝑔  ( J  pi1  Y ) )
7854, 64, 76, 77syl3anc 1218 . . . . . . . . . . 11  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( J  pi1  ( f `  0 ) ) 
~=ph𝑔  ( J  pi1  Y ) )
7953, 5gicen 15805 . . . . . . . . . . 11  |-  ( ( J  pi1  ( f `  0 ) )  ~=ph𝑔  ( J  pi1  Y )  ->  ( Base `  ( J  pi1  ( f ` 
0 ) ) ) 
~~  ( Base `  ( J  pi1  Y ) ) )
8078, 79syl 16 . . . . . . . . . 10  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( Base `  ( J  pi1 
( f `  0
) ) )  ~~  ( Base `  ( J  pi1  Y )
) )
81 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( Base `  ( J  pi1  Y ) )  ~~  1o )
82 entr 7361 . . . . . . . . . 10  |-  ( ( ( Base `  ( J  pi1  ( f `
 0 ) ) )  ~~  ( Base `  ( J  pi1  Y ) )  /\  ( Base `  ( J  pi1  Y )
)  ~~  1o )  ->  ( Base `  ( J  pi1  ( f `
 0 ) ) )  ~~  1o )
8380, 81, 82syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( Base `  ( J  pi1 
( f `  0
) ) )  ~~  1o )
84 en1eqsn 7542 . . . . . . . . 9  |-  ( ( [ ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ] (  ~=ph  `  J
)  e.  ( Base `  ( J  pi1 
( f `  0
) ) )  /\  ( Base `  ( J  pi1  ( f `  0 ) ) )  ~~  1o )  ->  ( Base `  ( J  pi1  ( f `
 0 ) ) )  =  { [
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) ] (  ~=ph  `  J ) } )
8575, 83, 84syl2anc 661 . . . . . . . 8  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( Base `  ( J  pi1 
( f `  0
) ) )  =  { [ ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) ] (  ~=ph  `  J ) } )
8668, 85eleqtrd 2519 . . . . . . 7  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  [ f ] (  ~=ph  `  J
)  e.  { [
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) ] (  ~=ph  `  J ) } )
87 elsni 3902 . . . . . . 7  |-  ( [ f ] (  ~=ph  `  J )  e.  { [ ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ] (  ~=ph  `  J
) }  ->  [ f ] (  ~=ph  `  J
)  =  [ ( ( 0 [,] 1
)  X.  { ( f `  0 ) } ) ] ( 
~=ph  `  J ) )
8886, 87syl 16 . . . . . 6  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  [ f ] (  ~=ph  `  J
)  =  [ ( ( 0 [,] 1
)  X.  { ( f `  0 ) } ) ] ( 
~=ph  `  J ) )
8913a1i 11 . . . . . . 7  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  (  ~=ph  `  J )  Er  (
II  Cn  J )
)
9089, 58erth 7145 . . . . . 6  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  ( f
(  ~=ph  `  J )
( ( 0 [,] 1 )  X.  {
( f `  0
) } )  <->  [ f ] (  ~=ph  `  J
)  =  [ ( ( 0 [,] 1
)  X.  { ( f `  0 ) } ) ] ( 
~=ph  `  J ) ) )
9188, 90mpbird 232 . . . . 5  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  (
f  e.  ( II 
Cn  J )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  ->  f (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) )
9291expr 615 . . . 4  |-  ( ( ( ( J  e. PCon  /\  Y  e.  X
)  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  /\  f  e.  ( II  Cn  J
) )  ->  (
( f `  0
)  =  ( f `
 1 )  -> 
f (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) )
9392ralrimiva 2799 . . 3  |-  ( ( ( J  e. PCon  /\  Y  e.  X )  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  ->  A. f  e.  ( II  Cn  J ) ( ( f ` 
0 )  =  ( f `  1 )  ->  f (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) ) )
94 isscon 27115 . . 3  |-  ( J  e. SCon 
<->  ( J  e. PCon  /\  A. f  e.  ( II 
Cn  J ) ( ( f `  0
)  =  ( f `
 1 )  -> 
f (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) ) )
9551, 93, 94sylanbrc 664 . 2  |-  ( ( ( J  e. PCon  /\  Y  e.  X )  /\  ( Base `  ( J  pi1  Y ) )  ~~  1o )  ->  J  e. SCon )
9650, 95impbida 828 1  |-  ( ( J  e. PCon  /\  Y  e.  X )  ->  ( J  e. SCon  <->  ( Base `  ( J  pi1  Y ) )  ~~  1o ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   {csn 3877   U.cuni 4091   class class class wbr 4292    X. cxp 4838   -->wf 5414   ` cfv 5418  (class class class)co 6091   1oc1o 6913    Er wer 7098   [cec 7099    ~~ cen 7307   0cc0 9282   1c1 9283   [,]cicc 11303   Basecbs 14174   0gc0g 14378   Grpcgrp 15410    ~=ph𝑔 cgic 15786   Topctop 18498  TopOnctopon 18499    Cn ccn 18828   IIcii 20451    ~=ph cphtpc 20541    pi1 cpi1 20575  PConcpcon 27108  SConcscon 27109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-ec 7103  df-qs 7107  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-icc 11307  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-divs 14447  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-grp 15545  df-mulg 15548  df-ghm 15745  df-gim 15787  df-gic 15788  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-cn 18831  df-cnp 18832  df-tx 19135  df-hmeo 19328  df-xms 19895  df-ms 19896  df-tms 19897  df-ii 20453  df-htpy 20542  df-phtpy 20543  df-phtpc 20564  df-pco 20577  df-om1 20578  df-pi1 20580  df-pcon 27110  df-scon 27111
This theorem is referenced by: (None)
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