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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 23483 | . . 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 7368 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ∈ 𝒫 ℂ) |
3 | 2 | elpwid 4508 | 1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 {crab 3110 ⊆ wss 3881 𝒫 cpw 4497 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 ℂcc 10524 < clt 10664 − cmin 10859 ℝ+crp 12377 abscabs 14585 –cn→ccncf 23481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-dm 5529 df-iota 6283 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-cncf 23483 |
This theorem is referenced by: cncff 23498 cncfi 23499 rescncf 23502 cncffvrn 23503 cncfco 23512 cncfcompt2 23513 cncfmpt2f 23520 cncfcnvcn 23530 cncombf 24262 cnlimci 24492 ulmcn 24994 efmul2picn 31977 mulcncff 42512 subcncff 42522 negcncfg 42523 addcncff 42526 ioccncflimc 42527 icocncflimc 42531 divcncff 42533 |
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