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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 23480 | . . 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 7382 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ∈ 𝒫 ℂ) |
3 | 2 | elpwid 4552 | 1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 {crab 3142 ⊆ wss 3935 𝒫 cpw 4538 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 ℂcc 10529 < clt 10669 − cmin 10864 ℝ+crp 12383 abscabs 14587 –cn→ccncf 23478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-xp 5555 df-dm 5559 df-iota 6308 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-cncf 23480 |
This theorem is referenced by: cncff 23495 cncfi 23496 rescncf 23499 cncffvrn 23500 cncfco 23509 cncfmpt2f 23516 cncfcnvcn 23523 cncombf 24253 cnlimci 24481 ulmcn 24981 efmul2picn 31862 mulcncff 42144 subcncff 42156 negcncfg 42157 addcncff 42160 ioccncflimc 42161 icocncflimc 42165 divcncff 42167 cncfcompt2 42175 |
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