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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 23935 | . 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 6781 | 1 ⊢ Rel (⇝𝑢‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1031 ∀wral 2896 ∃wrex 2897 Vcvv 3173 class class class wbr 4583 Rel wrel 5043 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℂcc 9813 < clt 9953 − cmin 10145 ℤcz 11254 ℤ≥cuz 11563 ℝ+crp 11708 abscabs 13822 ⇝𝑢culm 23934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ulm 23935 |
This theorem is referenced by: ulmval 23938 ulmdm 23951 ulmcau 23953 ulmdvlem3 23960 |
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