# Metamath Proof Explorer

## Theorem phtpy01

Description: Two path-homotopic paths have the same start and end point. (Contributed by Mario Carneiro, 23-Feb-2015)

Ref Expression
Hypotheses isphtpy.2
|- ( ph -> F e. ( II Cn J ) )
isphtpy.3
|- ( ph -> G e. ( II Cn J ) )
phtpyi.1
|- ( ph -> H e. ( F ( PHtpy ` J ) G ) )
Assertion phtpy01
|- ( ph -> ( ( F ` 0 ) = ( G ` 0 ) /\ ( F ` 1 ) = ( G ` 1 ) ) )

### Proof

Step Hyp Ref Expression
1 isphtpy.2
|-  ( ph -> F e. ( II Cn J ) )
2 isphtpy.3
|-  ( ph -> G e. ( II Cn J ) )
3 phtpyi.1
|-  ( ph -> H e. ( F ( PHtpy ` J ) G ) )
4 1elunit
|-  1 e. ( 0 [,] 1 )
5 1 2 3 phtpyi
|-  ( ( ph /\ 1 e. ( 0 [,] 1 ) ) -> ( ( 0 H 1 ) = ( F ` 0 ) /\ ( 1 H 1 ) = ( F ` 1 ) ) )
6 4 5 mpan2
|-  ( ph -> ( ( 0 H 1 ) = ( F ` 0 ) /\ ( 1 H 1 ) = ( F ` 1 ) ) )
7 6 simpld
|-  ( ph -> ( 0 H 1 ) = ( F ` 0 ) )
8 0elunit
|-  0 e. ( 0 [,] 1 )
9 iitopon
|-  II e. ( TopOn ` ( 0 [,] 1 ) )
10 9 a1i
|-  ( ph -> II e. ( TopOn ` ( 0 [,] 1 ) ) )
11 1 2 phtpyhtpy
|-  ( ph -> ( F ( PHtpy ` J ) G ) C_ ( F ( II Htpy J ) G ) )
12 11 3 sseldd
|-  ( ph -> H e. ( F ( II Htpy J ) G ) )
13 10 1 2 12 htpyi
|-  ( ( ph /\ 0 e. ( 0 [,] 1 ) ) -> ( ( 0 H 0 ) = ( F ` 0 ) /\ ( 0 H 1 ) = ( G ` 0 ) ) )
14 8 13 mpan2
|-  ( ph -> ( ( 0 H 0 ) = ( F ` 0 ) /\ ( 0 H 1 ) = ( G ` 0 ) ) )
15 14 simprd
|-  ( ph -> ( 0 H 1 ) = ( G ` 0 ) )
16 7 15 eqtr3d
|-  ( ph -> ( F ` 0 ) = ( G ` 0 ) )
17 6 simprd
|-  ( ph -> ( 1 H 1 ) = ( F ` 1 ) )
18 10 1 2 12 htpyi
|-  ( ( ph /\ 1 e. ( 0 [,] 1 ) ) -> ( ( 1 H 0 ) = ( F ` 1 ) /\ ( 1 H 1 ) = ( G ` 1 ) ) )
19 4 18 mpan2
|-  ( ph -> ( ( 1 H 0 ) = ( F ` 1 ) /\ ( 1 H 1 ) = ( G ` 1 ) ) )
20 19 simprd
|-  ( ph -> ( 1 H 1 ) = ( G ` 1 ) )
21 17 20 eqtr3d
|-  ( ph -> ( F ` 1 ) = ( G ` 1 ) )
22 16 21 jca
|-  ( ph -> ( ( F ` 0 ) = ( G ` 0 ) /\ ( F ` 1 ) = ( G ` 1 ) ) )