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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 24904 | . . 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 4609 | 1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 ⊆ wss 3951 𝒫 cpw 4600 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 ℂcc 11153 < clt 11295 − cmin 11492 ℝ+crp 13034 abscabs 15273 –cn→ccncf 24902 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-cncf 24904 |
| This theorem is referenced by: cncff 24919 cncfi 24920 rescncf 24923 cncfcdm 24924 cncfco 24933 cncfcompt2 24934 cncfmpt2f 24941 cncfcnvcn 24952 mulcncf 25480 cncombf 25693 cnlimci 25924 ulmcn 26442 efmul2picn 34611 mulcncff 45885 subcncff 45895 negcncfg 45896 addcncff 45899 ioccncflimc 45900 icocncflimc 45904 divcncff 45906 |
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