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Theorem pi1val 18535
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 18506 . . . 4  |-  pi 1  =  ( j  e. 
Top ,  y  e.  U. j  |->  ( ( j 
Om 1  y ) 
/.s  (  ~=ph  `  j ) ) )
32a1i 10 . . 3  |-  ( ph  ->  pi 1  =  ( j  e.  Top , 
y  e.  U. j  |->  ( ( j  Om 1  y )  /.s  (  ~=ph  `  j ) ) ) )
4 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
j  =  J )
5 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
y  =  Y )
64, 5oveq12d 5876 . . . . 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 2333 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  Om 1 
y )  =  O )
94fveq2d 5529 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
(  ~=ph  `  j )  =  (  ~=ph  `  J
) )
108, 9oveq12d 5876 . . 3  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( j  Om 1  y )  /.s  (  ~=ph  `  j ) )  =  ( O  /.s  (  ~=ph  `  J ) ) )
11 unieq 3836 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
1211adantl 452 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  U. J )
13 pi1val.1 . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
14 toponuni 16665 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1513, 14syl 15 . . . . 5  |-  ( ph  ->  X  =  U. J
)
1615adantr 451 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  X  =  U. J )
1712, 16eqtr4d 2318 . . 3  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  X )
18 topontop 16664 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
1913, 18syl 15 . . 3  |-  ( ph  ->  J  e.  Top )
20 pi1val.2 . . 3  |-  ( ph  ->  Y  e.  X )
21 ovex 5883 . . . 4  |-  ( O 
/.s  (  ~=ph  `  J ) )  e.  _V
2221a1i 10 . . 3  |-  ( ph  ->  ( O  /.s  (  ~=ph  `  J ) )  e. 
_V )
233, 10, 17, 19, 20, 22ovmpt2dx 5974 . 2  |-  ( ph  ->  ( J  pi 1  Y )  =  ( O  /.s  (  ~=ph  `  J ) ) )
241, 23syl5eq 2327 1  |-  ( ph  ->  G  =  ( O 
/.s  (  ~=ph  `  J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   U.cuni 3827   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    /.s cqus 13408   Topctop 16631  TopOnctopon 16632    ~=ph cphtpc 18467    Om 1 comi 18499    pi 1 cpi1 18501
This theorem is referenced by:  pi1bas  18536  pi1addf  18545  pi1addval  18546  pi1grplem  18547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-topon 16639  df-pi1 18506
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