Theorem sconnpht 31211

Description: A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
sconnpht	|- ( ( J e. SConn /\ F e. ( II Cn J ) /\ ( F ` 0 ) = ( F ` 1 ) ) -> F ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( F ` 0 ) } ) )

Proof of Theorem sconnpht

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		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 ) } ) ) ) )
2		fveq1 6190	. . . . . 6 |- ( f = F -> ( f ` 0 ) = ( F ` 0 ) )
3		fveq1 6190	. . . . . 6 |- ( f = F -> ( f ` 1 ) = ( F ` 1 ) )
4	2, 3	eqeq12d 2637	. . . . 5 |- ( f = F -> ( ( f ` 0 ) = ( f ` 1 ) <-> ( F ` 0 ) = ( F ` 1 ) ) )
5		id 22	. . . . . 6 |- ( f = F -> f = F )
6	2	sneqd 4189	. . . . . . 7 |- ( f = F -> { ( f ` 0 ) } = { ( F ` 0 ) } )
7	6	xpeq2d 5139	. . . . . 6 |- ( f = F -> ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) = ( ( 0 [,] 1 ) X. { ( F ` 0 ) } ) )
8	5, 7	breq12d 4666	. . . . 5 |- ( f = F -> ( f ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) <-> F ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( F ` 0 ) } ) ) )
9	4, 8	imbi12d 334	. . . 4 |- ( f = F -> ( ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) <-> ( ( F ` 0 ) = ( F ` 1 ) -> F ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( F ` 0 ) } ) ) ) )
10	9	rspccv 3306	. . 3 |- ( A. f e. ( II Cn J ) ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) -> ( F e. ( II Cn J ) -> ( ( F ` 0 ) = ( F ` 1 ) -> F ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( F ` 0 ) } ) ) ) )
11	1, 10	simplbiim 659	. 2 |- ( J e. SConn -> ( F e. ( II Cn J ) -> ( ( F ` 0 ) = ( F ` 1 ) -> F ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( F ` 0 ) } ) ) ) )
12	11	3imp 1256	1 |- ( ( J e. SConn /\ F e. ( II Cn J ) /\ ( F ` 0 ) = ( F ` 1 ) ) -> F ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( F ` 0 ) } ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {csn 4177   class class class wbr 4653   X. cxp 5112   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   [,]cicc 12178   Cn ccn 21028   IIcii 22678   ~=ph cphtpc 22768  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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-iota 5851  df-fv 5896  df-ov 6653  df-sconn 31204

This theorem is referenced by:  sconnpht2  31220  sconnpi1  31221  txsconn  31223