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Theorem tusval 20504
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 20496 . . 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 4255 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  U. u  =  U. U )
54dmeqd 5203 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  dom  U. u  =  dom  U. U )
65opeq2d 4220 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  <. ( Base `  ndx ) ,  dom  U. u >.  = 
<. ( Base `  ndx ) ,  dom  U. U >. )
73opeq2d 4220 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  <. ( UnifSet
`  ndx ) ,  u >.  =  <. ( UnifSet `  ndx ) ,  U >. )
86, 7preq12d 4114 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  { <. (
Base `  ndx ) ,  dom  U. u >. , 
<. ( UnifSet `  ndx ) ,  u >. }  =  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } )
93fveq2d 5868 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  (unifTop `  u )  =  (unifTop `  U ) )
109opeq2d 4220 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  <. (TopSet ` 
ndx ) ,  (unifTop `  u ) >.  =  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. )
118, 10oveq12d 6300 . 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 20462 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
13 ovex 6307 . . 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 5953 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 1379    e. wcel 1767   _Vcvv 3113   {cpr 4029   <.cop 4033   U.cuni 4245    |-> cmpt 4505   dom cdm 4999   ran crn 5000   ` cfv 5586  (class class class)co 6282   ndxcnx 14483   sSet csts 14484   Basecbs 14486  TopSetcts 14557   UnifSetcunif 14561  UnifOncust 20437  unifTopcutop 20468  toUnifSpctus 20493
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-csb 3436  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-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-ov 6285  df-ust 20438  df-tus 20496
This theorem is referenced by:  tuslem  20505
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