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Theorem pi1val 18934
Description: The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
Hypotheses
Ref Expression
pi1val.g  |-  G  =  ( J  pi 1  Y )
pi1val.1  |-  ( ph  ->  J  e.  (TopOn `  X ) )
pi1val.2  |-  ( ph  ->  Y  e.  X )
pi1val.o  |-  O  =  ( J  Om 1  Y )
Assertion
Ref Expression
pi1val  |-  ( ph  ->  G  =  ( O 
/.s  (  ~=ph  `  J ) ) )

Proof of Theorem pi1val
Dummy variables  j 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pi1val.g . 2  |-  G  =  ( J  pi 1  Y )
2 df-pi1 18905 . . . 4  |-  pi 1  =  ( j  e. 
Top ,  y  e.  U. j  |->  ( ( j 
Om 1  y ) 
/.s  (  ~=ph  `  j ) ) )
32a1i 11 . . 3  |-  ( ph  ->  pi 1  =  ( j  e.  Top , 
y  e.  U. j  |->  ( ( j  Om 1  y )  /.s  (  ~=ph  `  j ) ) ) )
4 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
j  =  J )
5 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
y  =  Y )
64, 5oveq12d 6039 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  Om 1 
y )  =  ( J  Om 1  Y
) )
7 pi1val.o . . . . 5  |-  O  =  ( J  Om 1  Y )
86, 7syl6eqr 2438 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  Om 1 
y )  =  O )
94fveq2d 5673 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
(  ~=ph  `  j )  =  (  ~=ph  `  J
) )
108, 9oveq12d 6039 . . 3  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( j  Om 1  y )  /.s  (  ~=ph  `  j ) )  =  ( O  /.s  (  ~=ph  `  J ) ) )
11 unieq 3967 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
1211adantl 453 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  U. J )
13 pi1val.1 . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
14 toponuni 16916 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1513, 14syl 16 . . . . 5  |-  ( ph  ->  X  =  U. J
)
1615adantr 452 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  X  =  U. J )
1712, 16eqtr4d 2423 . . 3  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  X )
18 topontop 16915 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
1913, 18syl 16 . . 3  |-  ( ph  ->  J  e.  Top )
20 pi1val.2 . . 3  |-  ( ph  ->  Y  e.  X )
21 ovex 6046 . . . 4  |-  ( O 
/.s  (  ~=ph  `  J ) )  e.  _V
2221a1i 11 . . 3  |-  ( ph  ->  ( O  /.s  (  ~=ph  `  J ) )  e. 
_V )
233, 10, 17, 19, 20, 22ovmpt2dx 6140 . 2  |-  ( ph  ->  ( J  pi 1  Y )  =  ( O  /.s  (  ~=ph  `  J ) ) )
241, 23syl5eq 2432 1  |-  ( ph  ->  G  =  ( O 
/.s  (  ~=ph  `  J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900   U.cuni 3958   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023    /.s cqus 13659   Topctop 16882  TopOnctopon 16883    ~=ph cphtpc 18866    Om 1 comi 18898    pi 1 cpi1 18900
This theorem is referenced by:  pi1bas  18935  pi1addf  18944  pi1addval  18945  pi1grplem  18946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-topon 16890  df-pi1 18905
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