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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 24918 | . . 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 7675 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ∈ 𝒫 ℂ) |
3 | 2 | elpwid 4614 | 1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 {crab 3433 ⊆ wss 3963 𝒫 cpw 4605 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℂcc 11151 < clt 11293 − cmin 11490 ℝ+crp 13032 abscabs 15270 –cn→ccncf 24916 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-dm 5699 df-iota 6516 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-cncf 24918 |
This theorem is referenced by: cncff 24933 cncfi 24934 rescncf 24937 cncfcdm 24938 cncfco 24947 cncfcompt2 24948 cncfmpt2f 24955 cncfcnvcn 24966 mulcncf 25494 cncombf 25707 cnlimci 25939 ulmcn 26457 efmul2picn 34590 mulcncff 45826 subcncff 45836 negcncfg 45837 addcncff 45840 ioccncflimc 45841 icocncflimc 45845 divcncff 45847 |
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