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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 23051 | . . 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 | elmpt2cl1 7137 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ∈ 𝒫 ℂ) |
3 | 2 | elpwid 4390 | 1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 ∀wral 3117 ∃wrex 3118 {crab 3121 ⊆ wss 3798 𝒫 cpw 4378 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 ↑𝑚 cmap 8122 ℂcc 10250 < clt 10391 − cmin 10585 ℝ+crp 12112 abscabs 14351 –cn→ccncf 23049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-xp 5348 df-dm 5352 df-iota 6086 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-cncf 23051 |
This theorem is referenced by: cncff 23066 cncfi 23067 rescncf 23070 cncffvrn 23071 cncfco 23080 cncfmpt2f 23087 cncfcnvcn 23094 cncombf 23824 cnlimci 24052 ulmcn 24552 efmul2picn 31212 mulcncff 40869 subcncff 40881 negcncfg 40882 addcncff 40885 ioccncflimc 40886 icocncflimc 40890 divcncff 40892 cncfcompt2 40900 |
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