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Theorem sconpi1 28881
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 28870 . . . . . . . . 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 2457 . . . . . . . . 9  |-  ( J  pi1  Y )  =  ( J  pi1  Y )
5 eqid 2457 . . . . . . . . 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 19561 . . . . . . . . . 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 21671 . . . . . . . 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 21621 . . . . . . . . . . . . 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 760 . . . . . . . . . . . . . 14  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  J  e. SCon )
16 simplr 755 . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . 15  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  ( f `  0 )  =  Y )
18 simprr 757 . . . . . . . . . . . . . . 15  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  ( f `  1 )  =  Y )
1917, 18eqtr4d 2501 . . . . . . . . . . . . . 14  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  ( f `  0 )  =  ( f `  1
) )
20 sconpht 28871 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 4044 . . . . . . . . . . . . . 14  |-  ( ( ( ( Y  e.  X  /\  J  e. SCon
)  /\  f  e.  ( II  Cn  J
) )  /\  (
( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) )  ->  { (
f `  0 ) }  =  { Y } )
2322xpeq2d 5032 . . . . . . . . . . . . 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 4480 . . . . . . . . . . . 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 7376 . . . . . . . . . . 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 2457 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 )  X.  { Y }
)  =  ( ( 0 [,] 1 )  X.  { Y }
)
284, 27pi1id 21677 . . . . . . . . . . . . 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 2498 . . . . . . . . . 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 4046 . . . . . . . . . . 11  |-  ( x  e.  { ( 0g
`  ( J  pi1  Y ) ) }  <->  x  =  ( 0g `  ( J  pi1  Y ) ) )
33 eqeq1 2461 . . . . . . . . . . 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 2949 . . . . . . 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 3505 . . . . 5  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  (
Base `  ( J  pi1  Y )
)  C_  { ( 0g `  ( J  pi1  Y ) ) } )
404pi1grp 21676 . . . . . . . 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 2457 . . . . . . . 8  |-  ( 0g
`  ( J  pi1  Y ) )  =  ( 0g `  ( J  pi1  Y ) )
435, 42grpidcl 16205 . . . . . . 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 4177 . . . . 5  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  { ( 0g `  ( J  pi1  Y ) ) }  C_  ( Base `  ( J  pi1  Y ) ) )
4639, 45eqssd 3516 . . . 4  |-  ( ( Y  e.  X  /\  J  e. SCon )  ->  (
Base `  ( J  pi1  Y )
)  =  { ( 0g `  ( J  pi1  Y ) ) } )
47 fvex 5882 . . . . 5  |-  ( 0g
`  ( J  pi1  Y ) )  e.  _V
4847ensn1 7598 . . . 4  |-  { ( 0g `  ( J  pi1  Y ) ) }  ~~  1o
4946, 48syl6eqbr 4493 . . 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 2457 . . . . . . . . 9  |-  ( J  pi1  ( f `
 0 ) )  =  ( J  pi1  ( f ` 
0 ) )
53 eqid 2457 . . . . . . . . 9  |-  ( Base `  ( J  pi1 
( f `  0
) ) )  =  ( Base `  ( J  pi1  ( f `
 0 ) ) )
54 simplll 759 . . . . . . . . . . 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 28867 . . . . . . . . . . 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 756 . . . . . . . . . . 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 21511 . . . . . . . . . . . 12  |-  ( 0 [,] 1 )  = 
U. II
6059, 7cnf 19874 . . . . . . . . . . 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 11663 . . . . . . . . . 10  |-  0  e.  ( 0 [,] 1
)
63 ffvelrn 6030 . . . . . . . . . 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 2458 . . . . . . . . 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 757 . . . . . . . . . 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 2465 . . . . . . . . 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 21672 . . . . . . . 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 2457 . . . . . . . . . . . . 13  |-  ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } )
7069pcoptcl 21647 . . . . . . . . . . . 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 1008 . . . . . . . . . 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 1009 . . . . . . . . . 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 1010 . . . . . . . . . 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 21672 . . . . . . . . 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 760 . . . . . . . . . . . 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 28879 . . . . . . . . . . . 12  |-  ( ( J  e. PCon  /\  (
f `  0 )  e.  X  /\  Y  e.  X )  ->  ( J  pi1  ( f `
 0 ) ) 
~=g𝑔 
( J  pi1  Y ) )
7854, 64, 76, 77syl3anc 1228 . . . . . . . . . . 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 ) ) 
~=g𝑔 
( J  pi1  Y ) )
7953, 5gicen 16452 . . . . . . . . . . 11  |-  ( ( J  pi1  ( f `  0 ) )  ~=g𝑔  ( 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 755 . . . . . . . . . 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 7586 . . . . . . . . . 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 7768 . . . . . . . . 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 2547 . . . . . . 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 4057 . . . . . . 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 7374 . . . . . 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 2871 . . 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 28868 . . 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 832 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {csn 4032   U.cuni 4251   class class class wbr 4456    X. cxp 5006   -->wf 5590   ` cfv 5594  (class class class)co 6296   1oc1o 7141    Er wer 7326   [cec 7327    ~~ cen 7532   0cc0 9509   1c1 9510   [,]cicc 11557   Basecbs 14644   0gc0g 14857   Grpcgrp 16180    ~=g𝑔 cgic 16433   Topctop 19521  TopOnctopon 19522    Cn ccn 19852   IIcii 21505    ~=ph cphtpc 21595    pi1 cpi1 21629  PConcpcon 28861  SConcscon 28862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-ec 7331  df-qs 7335  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-icc 11561  df-fz 11698  df-fzo 11822  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-qus 14926  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-grp 16184  df-mulg 16187  df-ghm 16392  df-gim 16434  df-gic 16435  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-cn 19855  df-cnp 19856  df-tx 20189  df-hmeo 20382  df-xms 20949  df-ms 20950  df-tms 20951  df-ii 21507  df-htpy 21596  df-phtpy 21597  df-phtpc 21618  df-pco 21631  df-om1 21632  df-pi1 21634  df-pcon 28863  df-scon 28864
This theorem is referenced by: (None)
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