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| Description: The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| ulmrel | ⊢ Rel (⇝𝑢‘𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ulm 26421 | . 2 ⊢ ⇝𝑢 = (𝑠 ∈ V ↦ {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}) | |
| 2 | 1 | relmptopab 7684 | 1 ⊢ Rel (⇝𝑢‘𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ w3a 1086 ∀wral 3060 ∃wrex 3069 Vcvv 3479 class class class wbr 5142 Rel wrel 5689 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 ℂcc 11154 < clt 11296 − cmin 11493 ℤcz 12615 ℤ≥cuz 12879 ℝ+crp 13035 abscabs 15274 ⇝𝑢culm 26420 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-ulm 26421 | 
| This theorem is referenced by: ulmval 26424 ulmdm 26437 ulmcau 26439 ulmdvlem3 26446 | 
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