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Theorem ulmrel 20247
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 20246 . 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 6251 1  |-  Rel  ( ~~> u `  S )
Colors of variables: wff set class
Syntax hints:    /\ w3a 936   A.wral 2666   E.wrex 2667   _Vcvv 2916   class class class wbr 4172   Rel wrel 4842   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   CCcc 8944    < clt 9076    - cmin 9247   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568   abscabs 11994   ~~> uculm 20245
This theorem is referenced by:  ulmval  20249  ulmdm  20262  ulmcau  20264  ulmdvlem3  20271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-ulm 20246
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