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| Mirrors > Home > MPE Home > Th. List > cncfrss2 | Structured version Visualization version GIF version | ||
| Description: Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.) |
| Ref | Expression |
|---|---|
| cncfrss2 | ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-cncf 24777 | . . 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 | elmpocl2 7634 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ∈ 𝒫 ℂ) |
| 3 | 2 | elpwid 4574 | 1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 {crab 3408 ⊆ wss 3916 𝒫 cpw 4565 class class class wbr 5109 ‘cfv 6513 (class class class)co 7389 ↑m cmap 8801 ℂcc 11072 < clt 11214 − cmin 11411 ℝ+crp 12957 abscabs 15206 –cn→ccncf 24775 |
| 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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pr 5389 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3919 df-un 3921 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-opab 5172 df-xp 5646 df-dm 5650 df-iota 6466 df-fv 6521 df-ov 7392 df-oprab 7393 df-mpo 7394 df-cncf 24777 |
| This theorem is referenced by: cncff 24792 cncfi 24793 rescncf 24796 climcncf 24799 cncfco 24806 cncfcnvcn 24825 cnlimci 25796 cncfmptssg 45862 cncfcompt 45874 |
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