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Theorem kqval 20312
Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
Assertion
Ref Expression
kqval  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
Distinct variable groups:    x, y, J    x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem kqval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 topontop 19512 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2 id 22 . . . . 5  |-  ( j  =  J  ->  j  =  J )
3 unieq 4171 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
4 rabeq 3028 . . . . . 6  |-  ( j  =  J  ->  { y  e.  j  |  x  e.  y }  =  { y  e.  J  |  x  e.  y } )
53, 4mpteq12dv 4445 . . . . 5  |-  ( j  =  J  ->  (
x  e.  U. j  |->  { y  e.  j  |  x  e.  y } )  =  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y } ) )
62, 5oveq12d 6214 . . . 4  |-  ( j  =  J  ->  (
j qTop  ( x  e. 
U. j  |->  { y  e.  j  |  x  e.  y } ) )  =  ( J qTop  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y } ) ) )
7 df-kq 20280 . . . 4  |- KQ  =  ( j  e.  Top  |->  ( j qTop  ( x  e. 
U. j  |->  { y  e.  j  |  x  e.  y } ) ) )
8 ovex 6224 . . . 4  |-  ( J qTop  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y } ) )  e.  _V
96, 7, 8fvmpt 5857 . . 3  |-  ( J  e.  Top  ->  (KQ `  J )  =  ( J qTop  ( x  e. 
U. J  |->  { y  e.  J  |  x  e.  y } ) ) )
101, 9syl 16 . 2  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y } ) ) )
11 kqval.2 . . . 4  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
12 toponuni 19513 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1312mpteq1d 4448 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )  =  ( x  e. 
U. J  |->  { y  e.  J  |  x  e.  y } ) )
1411, 13syl5eq 2435 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  F  =  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y } ) )
1514oveq2d 6212 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( J qTop  F )  =  ( J qTop  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y } ) ) )
1610, 15eqtr4d 2426 1  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826   {crab 2736   U.cuni 4163    |-> cmpt 4425   ` cfv 5496  (class class class)co 6196   qTop cqtop 14910   Topctop 19479  TopOnctopon 19480  KQckq 20279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-topon 19487  df-kq 20280
This theorem is referenced by:  kqtopon  20313  kqid  20314  kqopn  20320  kqcld  20321  t0kq  20404
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