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Mirrors > Home > MPE Home > Th. List > cncfrss2 | Structured version Visualization version GIF version |
Description: Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.) |
Ref | Expression |
---|---|
cncfrss2 | ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ 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 | elmpocl2 7664 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ∈ 𝒫 ℂ) |
3 | 2 | elpwid 4613 | 1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ 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 climcncf 24864 cncfco 24871 cncfcnvcn 24890 cnlimci 25862 cncfmptssg 45397 cncfcompt 45409 |
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