MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tusval Structured version   Unicode version

Theorem tusval 20639
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 20631 . . 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 4241 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  U. u  =  U. U )
54dmeqd 5192 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  dom  U. u  =  dom  U. U )
65opeq2d 4206 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  <. ( Base `  ndx ) ,  dom  U. u >.  = 
<. ( Base `  ndx ) ,  dom  U. U >. )
73opeq2d 4206 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  <. ( UnifSet
`  ndx ) ,  u >.  =  <. ( UnifSet `  ndx ) ,  U >. )
86, 7preq12d 4099 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  { <. (
Base `  ndx ) ,  dom  U. u >. , 
<. ( UnifSet `  ndx ) ,  u >. }  =  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } )
93fveq2d 5857 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  (unifTop `  u )  =  (unifTop `  U ) )
109opeq2d 4206 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  <. (TopSet ` 
ndx ) ,  (unifTop `  u ) >.  =  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. )
118, 10oveq12d 6296 . 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 20597 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
13 ovex 6306 . . 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 5943 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 1381    e. wcel 1802   _Vcvv 3093   {cpr 4013   <.cop 4017   U.cuni 4231    |-> cmpt 4492   dom cdm 4986   ran crn 4987   ` cfv 5575  (class class class)co 6278   ndxcnx 14503   sSet csts 14504   Basecbs 14506  TopSetcts 14577   UnifSetcunif 14581  UnifOncust 20572  unifTopcutop 20603  toUnifSpctus 20628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-iota 5538  df-fun 5577  df-fn 5578  df-fv 5583  df-ov 6281  df-ust 20573  df-tus 20631
This theorem is referenced by:  tuslem  20640
  Copyright terms: Public domain W3C validator