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Theorem ulmrel 22504
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 22503 . 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 6505 1  |-  Rel  ( ~~> u `  S )
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 973   A.wral 2814   E.wrex 2815   _Vcvv 3113   class class class wbr 4447   Rel wrel 5004   -->wf 5582   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   CCcc 9486    < clt 9624    - cmin 9801   ZZcz 10860   ZZ>=cuz 11078   RR+crp 11216   abscabs 13024   ~~> uculm 22502
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
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-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-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594  df-ulm 22503
This theorem is referenced by:  ulmval  22506  ulmdm  22519  ulmcau  22521  ulmdvlem3  22528
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