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Theorem kqval 19962
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 19194 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2 id 22 . . . . 5  |-  ( j  =  J  ->  j  =  J )
3 unieq 4253 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
4 rabeq 3107 . . . . . 6  |-  ( j  =  J  ->  { y  e.  j  |  x  e.  y }  =  { y  e.  J  |  x  e.  y } )
53, 4mpteq12dv 4525 . . . . 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 6300 . . . 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 19930 . . . 4  |- KQ  =  ( j  e.  Top  |->  ( j qTop  ( x  e. 
U. j  |->  { y  e.  j  |  x  e.  y } ) ) )
8 ovex 6307 . . . 4  |-  ( J qTop  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y } ) )  e.  _V
96, 7, 8fvmpt 5948 . . 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 19195 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1312mpteq1d 4528 . . . 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 2520 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  F  =  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y } ) )
1514oveq2d 6298 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( J qTop  F )  =  ( J qTop  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y } ) ) )
1610, 15eqtr4d 2511 1  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {crab 2818   U.cuni 4245    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282   qTop cqtop 14754   Topctop 19161  TopOnctopon 19162  KQckq 19929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-topon 19169  df-kq 19930
This theorem is referenced by:  kqtopon  19963  kqid  19964  kqopn  19970  kqcld  19971  t0kq  20054
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