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Theorem ulmrel 23412
Description: The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
ulmrel  |-  Rel  ( ~~> u `  S )

Proof of Theorem ulmrel
Dummy variables  f 
j  k  n  s  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ulm 23411 . 2  |-  ~~> u  =  ( s  e.  _V  |->  { <. f ,  y
>.  |  E. n  e.  ZZ  ( f : ( ZZ>= `  n ) --> ( CC  ^m  s
)  /\  y :
s --> CC  /\  A. x  e.  RR+  E. j  e.  ( ZZ>= `  n ) A. k  e.  ( ZZ>=
`  j ) A. z  e.  s  ( abs `  ( ( ( f `  k ) `
 z )  -  ( y `  z
) ) )  < 
x ) } )
21relmptopab 6536 1  |-  Rel  ( ~~> u `  S )
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 1007   A.wral 2756   E.wrex 2757   _Vcvv 3031   class class class wbr 4395   Rel wrel 4844   -->wf 5585   ` cfv 5589  (class class class)co 6308    ^m cmap 7490   CCcc 9555    < clt 9693    - cmin 9880   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   abscabs 13374   ~~> uculm 23410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-ulm 23411
This theorem is referenced by:  ulmval  23414  ulmdm  23427  ulmcau  23429  ulmdvlem3  23436
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