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Mirrors > Home > MPE Home > Th. List > cncfrss | Structured version Visualization version GIF version |
Description: Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.) |
Ref | Expression |
---|---|
cncfrss | ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cncf 24842 | . . 3 ⊢ –cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏 ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑎 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)}) | |
2 | 1 | elmpocl1 7663 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ∈ 𝒫 ℂ) |
3 | 2 | elpwid 4613 | 1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 {crab 3418 ⊆ wss 3944 𝒫 cpw 4604 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 ↑m cmap 8845 ℂcc 11138 < clt 11280 − cmin 11476 ℝ+crp 13009 abscabs 15217 –cn→ccncf 24840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5684 df-dm 5688 df-iota 6501 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-cncf 24842 |
This theorem is referenced by: cncff 24857 cncfi 24858 rescncf 24861 cncfcdm 24862 cncfco 24871 cncfcompt2 24872 cncfmpt2f 24879 cncfcnvcn 24890 mulcncf 25418 cncombf 25631 cnlimci 25862 ulmcn 26380 efmul2picn 34359 mulcncff 45396 subcncff 45406 negcncfg 45407 addcncff 45410 ioccncflimc 45411 icocncflimc 45415 divcncff 45417 |
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