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Mirrors > Home > MPE Home > Th. List > ulmrel | Structured version Visualization version GIF version |
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 24892 | . 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 7384 | 1 ⊢ Rel (⇝𝑢‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1079 ∀wral 3135 ∃wrex 3136 Vcvv 3492 class class class wbr 5057 Rel wrel 5553 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 ℂcc 10523 < clt 10663 − cmin 10858 ℤcz 11969 ℤ≥cuz 12231 ℝ+crp 12377 abscabs 14581 ⇝𝑢culm 24891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-ulm 24892 |
This theorem is referenced by: ulmval 24895 ulmdm 24908 ulmcau 24910 ulmdvlem3 24917 |
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