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Theorem ulmrel 25753
Description: The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
ulmrel Rel (β‡π‘’β€˜π‘†)

Proof of Theorem ulmrel
Dummy variables 𝑓 𝑗 π‘˜ 𝑛 𝑠 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ulm 25752 . 2 ⇝𝑒 = (𝑠 ∈ V ↦ {βŸ¨π‘“, π‘¦βŸ© ∣ βˆƒπ‘› ∈ β„€ (𝑓:(β„€β‰₯β€˜π‘›)⟢(β„‚ ↑m 𝑠) ∧ 𝑦:π‘ βŸΆβ„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘— ∈ (β„€β‰₯β€˜π‘›)βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)βˆ€π‘§ ∈ 𝑠 (absβ€˜(((π‘“β€˜π‘˜)β€˜π‘§) βˆ’ (π‘¦β€˜π‘§))) < π‘₯)})
21relmptopab 7608 1 Rel (β‡π‘’β€˜π‘†)
Colors of variables: wff setvar class
Syntax hints:   ∧ w3a 1088  βˆ€wral 3065  βˆƒwrex 3074  Vcvv 3448   class class class wbr 5110  Rel wrel 5643  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362   ↑m cmap 8772  β„‚cc 11056   < clt 11196   βˆ’ cmin 11392  β„€cz 12506  β„€β‰₯cuz 12770  β„+crp 12922  abscabs 15126  β‡π‘’culm 25751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fv 6509  df-ulm 25752
This theorem is referenced by:  ulmval  25755  ulmdm  25768  ulmcau  25770  ulmdvlem3  25777
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