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Theorem ulmrel 19855
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 19854 . 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 6149 1  |-  Rel  ( ~~> u `  S )
Colors of variables: wff set class
Syntax hints:    /\ w3a 934   A.wral 2619   E.wrex 2620   _Vcvv 2864   class class class wbr 4102   Rel wrel 4773   -->wf 5330   ` cfv 5334  (class class class)co 5942    ^m cmap 6857   CCcc 8822    < clt 8954    - cmin 9124   ZZcz 10113   ZZ>=cuz 10319   RR+crp 10443   abscabs 11809   ~~> uculm 19853
This theorem is referenced by:  ulmval  19857  ulmdm  19870  ulmcau  19872  ulmdvlem3  19879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fv 5342  df-ulm 19854
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