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Theorem tusval 19966
Description: The value of the uniform space mapping function. (Contributed by Thierry Arnoux, 5-Dec-2017.)
Assertion
Ref Expression
tusval  |-  ( U  e.  (UnifOn `  X
)  ->  (toUnifSp `  U
)  =  ( {
<. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )

Proof of Theorem tusval
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 df-tus 19958 . . 3  |- toUnifSp  =  ( u  e.  U. ran UnifOn  |->  ( { <. ( Base `  ndx ) ,  dom  U. u >. ,  <. ( UnifSet `  ndx ) ,  u >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  u ) >. ) )
21a1i 11 . 2  |-  ( U  e.  (UnifOn `  X
)  -> toUnifSp  =  ( u  e.  U. ran UnifOn  |->  ( {
<. ( Base `  ndx ) ,  dom  U. u >. ,  <. ( UnifSet `  ndx ) ,  u >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  u ) >. ) ) )
3 simpr 461 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  u  =  U )
43unieqd 4202 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  U. u  =  U. U )
54dmeqd 5143 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  dom  U. u  =  dom  U. U )
65opeq2d 4167 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  <. ( Base `  ndx ) ,  dom  U. u >.  = 
<. ( Base `  ndx ) ,  dom  U. U >. )
73opeq2d 4167 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  <. ( UnifSet
`  ndx ) ,  u >.  =  <. ( UnifSet `  ndx ) ,  U >. )
86, 7preq12d 4063 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  { <. (
Base `  ndx ) ,  dom  U. u >. , 
<. ( UnifSet `  ndx ) ,  u >. }  =  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } )
93fveq2d 5796 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  (unifTop `  u )  =  (unifTop `  U ) )
109opeq2d 4167 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  <. (TopSet ` 
ndx ) ,  (unifTop `  u ) >.  =  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. )
118, 10oveq12d 6211 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  ( { <. ( Base `  ndx ) ,  dom  U. u >. ,  <. ( UnifSet `  ndx ) ,  u >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  u ) >. )  =  ( {
<. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
12 elrnust 19924 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
13 ovex 6218 . . 3  |-  ( {
<. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. )  e.  _V
1413a1i 11 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. )  e.  _V )
152, 11, 12, 14fvmptd 5881 1  |-  ( U  e.  (UnifOn `  X
)  ->  (toUnifSp `  U
)  =  ( {
<. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3071   {cpr 3980   <.cop 3984   U.cuni 4192    |-> cmpt 4451   dom cdm 4941   ran crn 4942   ` cfv 5519  (class class class)co 6193   ndxcnx 14282   sSet csts 14283   Basecbs 14285  TopSetcts 14355   UnifSetcunif 14359  UnifOncust 19899  unifTopcutop 19930  toUnifSpctus 19955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-fv 5527  df-ov 6196  df-ust 19900  df-tus 19958
This theorem is referenced by:  tuslem  19967
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