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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 23488 | . . 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 7391 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ∈ 𝒫 ℂ) |
3 | 2 | elpwid 4552 | 1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 {crab 3144 ⊆ wss 3938 𝒫 cpw 4541 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 ℂcc 10537 < clt 10677 − cmin 10872 ℝ+crp 12392 abscabs 14595 –cn→ccncf 23486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-dm 5567 df-iota 6316 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-cncf 23488 |
This theorem is referenced by: cncff 23503 cncfi 23504 rescncf 23507 climcncf 23510 cncfco 23517 cncfcnvcn 23531 cnlimci 24489 cncfmptssg 42160 cncfcompt 42173 |
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