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Theorem pi1val 20628
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 20599 . . . 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 755 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
j  =  J )
5 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
y  =  Y )
64, 5oveq12d 6128 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  Om1 
y )  =  ( J  Om1  Y ) )
7 pi1val.o . . . . 5  |-  O  =  ( J  Om1  Y )
86, 7syl6eqr 2493 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  Om1 
y )  =  O )
94fveq2d 5714 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
(  ~=ph  `  j )  =  (  ~=ph  `  J
) )
108, 9oveq12d 6128 . . 3  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( j  Om1  y )  /.s  (  ~=ph  `  j ) )  =  ( O  /.s  (  ~=ph  `  J ) ) )
11 unieq 4118 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
1211adantl 466 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  U. J )
13 pi1val.1 . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
14 toponuni 18551 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1513, 14syl 16 . . . . 5  |-  ( ph  ->  X  =  U. J
)
1615adantr 465 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  X  =  U. J )
1712, 16eqtr4d 2478 . . 3  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  X )
18 topontop 18550 . . . 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 6135 . . . 4  |-  ( O 
/.s  (  ~=ph  `  J ) )  e.  _V
2221a1i 11 . . 3  |-  ( ph  ->  ( O  /.s  (  ~=ph  `  J ) )  e. 
_V )
233, 10, 17, 19, 20, 22ovmpt2dx 6236 . 2  |-  ( ph  ->  ( J  pi1  Y )  =  ( O  /.s  (  ~=ph  `  J ) ) )
241, 23syl5eq 2487 1  |-  ( ph  ->  G  =  ( O 
/.s  (  ~=ph  `  J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2991   U.cuni 4110   ` cfv 5437  (class class class)co 6110    e. cmpt2 6112    /.s cqus 14462   Topctop 18517  TopOnctopon 18518    ~=ph cphtpc 20560    Om1 comi 20592    pi1 cpi1 20594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-iota 5400  df-fun 5439  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-topon 18525  df-pi1 20599
This theorem is referenced by:  pi1bas  20629  pi1addf  20638  pi1addval  20639  pi1grplem  20640
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