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Theorem sconnpi1 31221
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
sconnpi1.1 |- X = U. J
Assertion
Ref Expression
sconnpi1 |- ( ( J e. PConn /\ Y e. X ) -> ( J e. SConn <-> ( Base ` ( J pi1 Y ) ) ~~ 1o ) )
Proof of Theorem sconnpi1
Dummy variables x f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sconntop 31210 . . . . . . . . 9 |- ( J e. SConn -> J e. Top )
21adantl 482 . . . . . . . 8 |- ( ( Y e. X /\ J e. SConn ) -> J e. Top )
3 simpl 473 . . . . . . . 8 |- ( ( Y e. X /\ J e. SConn ) -> Y e. X )
4 eqid 2622 . . . . . . . . 9 |- ( J pi1 Y ) = ( J pi1 Y )
5 eqid 2622 . . . . . . . . 9 |- ( Base ` ( J pi1 Y ) ) = ( Base ` ( J pi1 Y ) )
6 simpl 473 . . . . . . . . . 10 |- ( ( J e. Top /\ Y e. X ) -> J e. Top )
7 sconnpi1.1 . . . . . . . . . . 11 |- X = U. J
87toptopon 20722 . . . . . . . . . 10 |- ( J e. Top <-> J e. (TopOn ` X ) )
96, 8sylib 208 . . . . . . . . 9 |- ( ( J e. Top /\ Y e. X ) -> J e. (TopOn ` X ) )
10 simpr 477 . . . . . . . . 9 |- ( ( J e. Top /\ Y e. X ) -> Y e. X )
114, 5, 9, 10elpi1 22845 . . . . . . . 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 693 . . . . . . 7 |- ( ( Y e. X /\ J e. SConn ) -> ( x e. ( Base ` ( J pi1 Y ) ) <-> E. f e. ( II Cn J ) ( ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) /\ x = [ f ] ( ~=ph ` J ) ) ) )
13 phtpcer 22794 . . . . . . . . . . . . 13 |- ( ~=ph ` J ) Er ( II Cn J )
1413a1i 11 . . . . . . . . . . . 12 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> ( ~=ph ` J ) Er ( II Cn J ) )
15 simpllr 799 . . . . . . . . . . . . . 14 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> J e. SConn )
16 simplr 792 . . . . . . . . . . . . . 14 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> f e. ( II Cn J ) )
17 simprl 794 . . . . . . . . . . . . . . 15 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> ( f ` 0 ) = Y )
18 simprr 796 . . . . . . . . . . . . . . 15 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> ( f ` 1 ) = Y )
1917, 18eqtr4d 2659 . . . . . . . . . . . . . 14 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> ( f ` 0 ) = ( f ` 1 ) )
20 sconnpht 31211 . . . . . . . . . . . . . 14 |- ( ( J e. SConn /\ f e. ( II Cn J ) /\ ( f ` 0 ) = ( f ` 1 ) ) -> f ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) )
2115, 16, 19, 20syl3anc 1326 . . . . . . . . . . . . 13 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> f ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) )
2217sneqd 4189 . . . . . . . . . . . . . 14 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> { ( f ` 0 ) } = { Y } )
2322xpeq2d 5139 . . . . . . . . . . . . 13 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) = ( ( 0 [,] 1 ) X. { Y } ) )
2421, 23breqtrd 4679 . . . . . . . . . . . 12 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> f ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { Y } ) )
2514, 24erthi 7793 . . . . . . . . . . 11 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> [ f ] ( ~=ph ` J ) = [ ( ( 0 [,] 1 ) X. { Y } ) ] ( ~=ph ` J ) )
262, 8sylib 208 . . . . . . . . . . . . 13 |- ( ( Y e. X /\ J e. SConn ) -> J e. (TopOn ` X ) )
27 eqid 2622 . . . . . . . . . . . . . 14 |- ( ( 0 [,] 1 ) X. { Y } ) = ( ( 0 [,] 1 ) X. { Y } )
284, 27pi1id 22851 . . . . . . . . . . . . 13 |- ( ( J e. (TopOn ` X ) /\ Y e. X ) -> [ ( ( 0 [,] 1 ) X. { Y } ) ] ( ~=ph ` J ) = ( 0g ` ( J pi1 Y ) ) )
2926, 3, 28syl2anc 693 . . . . . . . . . . . 12 |- ( ( Y e. X /\ J e. SConn ) -> [ ( ( 0 [,] 1 ) X. { Y } ) ] ( ~=ph ` J ) = ( 0g ` ( J pi1 Y ) ) )
3029ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> [ ( ( 0 [,] 1 ) X. { Y } ) ] ( ~=ph ` J ) = ( 0g ` ( J pi1 Y ) ) )
3125, 30eqtrd 2656 . . . . . . . . . 10 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> [ f ] ( ~=ph ` J ) = ( 0g ` ( J pi1 Y ) ) )
32 velsn 4193 . . . . . . . . . . 11 |- ( x e. { ( 0g ` ( J pi1 Y ) ) } <-> x = ( 0g ` ( J pi1 Y ) ) )
33 eqeq1 2626 . . . . . . . . . . 11 |- ( x = [ f ] ( ~=ph ` J ) -> ( x = ( 0g ` ( J pi1 Y ) ) <-> [ f ] ( ~=ph ` J ) = ( 0g ` ( J pi1 Y ) ) ) )
3432, 33syl5bb 272 . . . . . . . . . 10 |- ( x = [ f ] ( ~=ph ` J ) -> ( x e. { ( 0g ` ( J pi1 Y ) ) } <-> [ f ] ( ~=ph ` J ) = ( 0g ` ( J pi1 Y ) ) ) )
3531, 34syl5ibrcom 237 . . . . . . . . 9 |- ( ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) /\ ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) -> ( x = [ f ] ( ~=ph ` J ) -> x e. { ( 0g ` ( J pi1 Y ) ) } ) )
3635expimpd 629 . . . . . . . 8 |- ( ( ( Y e. X /\ J e. SConn ) /\ f e. ( II Cn J ) ) -> ( ( ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) /\ x = [ f ] ( ~=ph ` J ) ) -> x e. { ( 0g ` ( J pi1 Y ) ) } ) )
3736rexlimdva 3031 . . . . . . 7 |- ( ( Y e. X /\ J e. SConn ) -> ( E. f e. ( II Cn J ) ( ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) /\ x = [ f ] ( ~=ph ` J ) ) -> x e. { ( 0g ` ( J pi1 Y ) ) } ) )
3812, 37sylbid 230 . . . . . 6 |- ( ( Y e. X /\ J e. SConn ) -> ( x e. ( Base ` ( J pi1 Y ) ) -> x e. { ( 0g ` ( J pi1 Y ) ) } ) )
3938ssrdv 3609 . . . . 5 |- ( ( Y e. X /\ J e. SConn ) -> ( Base ` ( J pi1 Y ) ) C_ { ( 0g ` ( J pi1 Y ) ) } )
404pi1grp 22850 . . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ Y e. X ) -> ( J pi1 Y ) e. Grp )
4126, 3, 40syl2anc 693 . . . . . . 7 |- ( ( Y e. X /\ J e. SConn ) -> ( J pi1 Y ) e. Grp )
42 eqid 2622 . . . . . . . 8 |- ( 0g ` ( J pi1 Y ) ) = ( 0g ` ( J pi1 Y ) )
435, 42grpidcl 17450 . . . . . . 7 |- ( ( J pi1 Y ) e. Grp -> ( 0g ` ( J pi1 Y ) ) e. ( Base ` ( J pi1 Y ) ) )
4441, 43syl 17 . . . . . 6 |- ( ( Y e. X /\ J e. SConn ) -> ( 0g ` ( J pi1 Y ) ) e. ( Base ` ( J pi1 Y ) ) )
4544snssd 4340 . . . . 5 |- ( ( Y e. X /\ J e. SConn ) -> { ( 0g ` ( J pi1 Y ) ) } C_ ( Base ` ( J pi1 Y ) ) )
4639, 45eqssd 3620 . . . 4 |- ( ( Y e. X /\ J e. SConn ) -> ( Base ` ( J pi1 Y ) ) = { ( 0g ` ( J pi1 Y ) ) } )
47 fvex 6201 . . . . 5 |- ( 0g ` ( J pi1 Y ) ) e. _V
4847ensn1 8020 . . . 4 |- { ( 0g ` ( J pi1 Y ) ) } ~~ 1o
4946, 48syl6eqbr 4692 . . 3 |- ( ( Y e. X /\ J e. SConn ) -> ( Base ` ( J pi1 Y ) ) ~~ 1o )
5049adantll 750 . 2 |- ( ( ( J e. PConn /\ Y e. X ) /\ J e. SConn ) -> ( Base ` ( J pi1 Y ) ) ~~ 1o )
51 simpll 790 . . 3 |- ( ( ( J e. PConn /\ Y e. X ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) -> J e. PConn )
52 eqid 2622 . . . . . . . . 9 |- ( J pi1 ( f ` 0 ) ) = ( J pi1 ( f ` 0 ) )
53 eqid 2622 . . . . . . . . 9 |- ( Base ` ( J pi1 ( f ` 0 ) ) ) = ( Base ` ( J pi1 ( f ` 0 ) ) )
54 simplll 798 . . . . . . . . . . 11 |- ( ( ( ( J e. PConn /\ Y e. X ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) /\ ( f e. ( II Cn J ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> J e. PConn )
55 pconntop 31207 . . . . . . . . . . 11 |- ( J e. PConn -> J e. Top )
5654, 55syl 17 . . . . . . . . . 10 |- ( ( ( ( J e. PConn /\ Y e. X ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) /\ ( f e. ( II Cn J ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> J e. Top )
5756, 8sylib 208 . . . . . . . . 9 |- ( ( ( ( J e. PConn /\ Y e. X ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) /\ ( f e. ( II Cn J ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> J e. (TopOn ` X ) )
58 simprl 794 . . . . . . . . . . 11 |- ( ( ( ( J e. PConn /\ Y e. X ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) /\ ( f e. ( II Cn J ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> f e. ( II Cn J ) )
59 iiuni 22684 . . . . . . . . . . . 12 |- ( 0 [,] 1 ) = U. II
6059, 7cnf 21050 . . . . . . . . . . 11 |- ( f e. ( II Cn J ) -> f : ( 0 [,] 1 ) --> X )
6158, 60syl 17 . . . . . . . . . 10 |- ( ( ( ( J e. PConn /\ Y e. X ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) /\ ( f e. ( II Cn J ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> f : ( 0 [,] 1 ) --> X )
62 0elunit 12290 . . . . . . . . . 10 |- 0 e. ( 0 [,] 1 )
63 ffvelrn 6357 . . . . . . . . . 10 |- ( ( f : ( 0 [,] 1 ) --> X /\ 0 e. ( 0 [,] 1 ) ) -> ( f ` 0 ) e. X )
6461, 62, 63sylancl 694 . . . . . . . . 9 |- ( ( ( ( J e. PConn /\ Y e. X ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) /\ ( f e. ( II Cn J ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> ( f ` 0 ) e. X )
65 eqidd 2623 . . . . . . . . 9 |- ( ( ( ( J e. PConn /\ Y e. X ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) /\ ( f e. ( II Cn J ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> ( f ` 0 ) = ( f ` 0 ) )
66 simprr 796 . . . . . . . . . 10 |- ( ( ( ( J e. PConn /\ Y e. X ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) /\ ( f e. ( II Cn J ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> ( f ` 0 ) = ( f ` 1 ) )
6766eqcomd 2628 . . . . . . . . 9 |- ( ( ( ( J e. PConn /\ 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 22846 . . . . . . . 8 |- ( ( ( ( J e. PConn /\ 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 2622 . . . . . . . . . . . . 13 |- ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) = ( ( 0 [,] 1 ) X. { ( f ` 0 ) } )
7069pcoptcl 22821 . . . . . . . . . . . 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 693 . . . . . . . . . . 11 |- ( ( ( ( J e. PConn /\ 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 1073 . . . . . . . . . 10 |- ( ( ( ( J e. PConn /\ 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 1074 . . . . . . . . . 10 |- ( ( ( ( J e. PConn /\ 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 1075 . . . . . . . . . 10 |- ( ( ( ( J e. PConn /\ 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 22846 . . . . . . . . 9 |- ( ( ( ( J e. PConn /\ 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 799 . . . . . . . . . . . 12 |- ( ( ( ( J e. PConn /\ Y e. X ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) /\ ( f e. ( II Cn J ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> Y e. X )
777, 52, 4, 53, 5pconnpi1 31219 . . . . . . . . . . . 12 |- ( ( J e. PConn /\ ( f ` 0 ) e. X /\ Y e. X ) -> ( J pi1 ( f ` 0 ) ) ~=g𝑔 ( J pi1 Y ) )
7854, 64, 76, 77syl3anc 1326 . . . . . . . . . . 11 |- ( ( ( ( J e. PConn /\ 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 17720 . . . . . . . . . . 11 |- ( ( J pi1 ( f ` 0 ) ) ~=g𝑔 ( J pi1 Y ) -> ( Base ` ( J pi1 ( f ` 0 ) ) ) ~~ ( Base ` ( J pi1 Y ) ) )
8078, 79syl 17 . . . . . . . . . 10 |- ( ( ( ( J e. PConn /\ 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 792 . . . . . . . . . 10 |- ( ( ( ( J e. PConn /\ Y e. X ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) /\ ( f e. ( II Cn J ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> ( Base ` ( J pi1 Y ) ) ~~ 1o )
82 entr 8008 . . . . . . . . . 10 |- ( ( ( Base ` ( J pi1 ( f ` 0 ) ) ) ~~ ( Base ` ( J pi1 Y ) ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) -> ( Base ` ( J pi1 ( f ` 0 ) ) ) ~~ 1o )
8380, 81, 82syl2anc 693 . . . . . . . . 9 |- ( ( ( ( J e. PConn /\ 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 8190 . . . . . . . . 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 693 . . . . . . . 8 |- ( ( ( ( J e. PConn /\ 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 2703 . . . . . . 7 |- ( ( ( ( J e. PConn /\ 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 4194 . . . . . . 7 |- ( [ f ] ( ~=ph ` J ) e. { [ ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ] ( ~=ph ` J ) } -> [ f ] ( ~=ph ` J ) = [ ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ] ( ~=ph ` J ) )
8886, 87syl 17 . . . . . 6 |- ( ( ( ( J e. PConn /\ 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. PConn /\ 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 7791 . . . . . 6 |- ( ( ( ( J e. PConn /\ 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 247 . . . . 5 |- ( ( ( ( J e. PConn /\ 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 643 . . . 4 |- ( ( ( ( J e. PConn /\ 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 2966 . . 3 |- ( ( ( J e. PConn /\ 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 issconn 31208 . . 3 |- ( J e. SConn <-> ( J e. PConn /\ A. f e. ( II Cn J ) ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) ) )
9551, 93, 94sylanbrc 698 . 2 |- ( ( ( J e. PConn /\ Y e. X ) /\ ( Base ` ( J pi1 Y ) ) ~~ 1o ) -> J e. SConn )
9650, 95impbida 877 1 |- ( ( J e. PConn /\ Y e. X ) -> ( J e. SConn <-> ( Base ` ( J pi1 Y ) ) ~~ 1o ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {csn 4177   U.cuni 4436   class class class wbr 4653   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   1oc1o 7553   Er wer 7739   [cec 7740   ~~ cen 7952   0cc0 9936   1c1 9937   [,]cicc 12178   Basecbs 15857   0gc0g 16100   Grpcgrp 17422   ~=g𝑔 cgic 17700   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   IIcii 22678   ~=ph cphtpc 22768   pi1 cpi1 22803  PConncpconn 31201  SConncsconn 31202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-qus 16169  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-mulg 17541  df-ghm 17658  df-gim 17701  df-gic 17702  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-ii 22680  df-htpy 22769  df-phtpy 22770  df-phtpc 22791  df-pco 22805  df-om1 22806  df-pi1 22808  df-pconn 31203  df-sconn 31204
This theorem is referenced by: (None)
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