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Theorem pi1val 21703
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  pi1  Y )
pi1val.1  |-  ( ph  ->  J  e.  (TopOn `  X ) )
pi1val.2  |-  ( ph  ->  Y  e.  X )
pi1val.o  |-  O  =  ( J  Om1  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  pi1  Y )
2 df-pi1 21674 . . . 4  |-  pi1 
=  ( j  e. 
Top ,  y  e.  U. j  |->  ( ( j 
Om1  y ) 
/.s  (  ~=ph  `  j ) ) )
32a1i 11 . . 3  |-  ( ph  ->  pi1  =  ( j  e.  Top , 
y  e.  U. j  |->  ( ( j  Om1  y )  /.s  (  ~=ph  `  j ) ) ) )
4 simprl 754 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
j  =  J )
5 simprr 755 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
y  =  Y )
64, 5oveq12d 6288 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  Om1 
y )  =  ( J  Om1  Y ) )
7 pi1val.o . . . . 5  |-  O  =  ( J  Om1  Y )
86, 7syl6eqr 2513 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  Om1 
y )  =  O )
94fveq2d 5852 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
(  ~=ph  `  j )  =  (  ~=ph  `  J
) )
108, 9oveq12d 6288 . . 3  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( j  Om1  y )  /.s  (  ~=ph  `  j ) )  =  ( O  /.s  (  ~=ph  `  J ) ) )
11 unieq 4243 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
1211adantl 464 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  U. J )
13 pi1val.1 . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
14 toponuni 19595 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1513, 14syl 16 . . . . 5  |-  ( ph  ->  X  =  U. J
)
1615adantr 463 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  X  =  U. J )
1712, 16eqtr4d 2498 . . 3  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  X )
18 topontop 19594 . . . 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 6298 . . . 4  |-  ( O 
/.s  (  ~=ph  `  J ) )  e.  _V
2221a1i 11 . . 3  |-  ( ph  ->  ( O  /.s  (  ~=ph  `  J ) )  e. 
_V )
233, 10, 17, 19, 20, 22ovmpt2dx 6402 . 2  |-  ( ph  ->  ( J  pi1  Y )  =  ( O  /.s  (  ~=ph  `  J ) ) )
241, 23syl5eq 2507 1  |-  ( ph  ->  G  =  ( O 
/.s  (  ~=ph  `  J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   U.cuni 4235   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272    /.s cqus 14994   Topctop 19561  TopOnctopon 19562    ~=ph cphtpc 21635    Om1 comi 21667    pi1 cpi1 21669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-topon 19569  df-pi1 21674
This theorem is referenced by:  pi1bas  21704  pi1addf  21713  pi1addval  21714  pi1grplem  21715
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