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Mirrors > Home > ILE Home > Th. List > cncfrss2 | 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 12716 | . . 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 5963 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ∈ 𝒫 ℂ) |
3 | 2 | elpwid 3516 | 1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ∀wral 2414 ∃wrex 2415 {crab 2418 ⊆ wss 3066 𝒫 cpw 3505 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 ↑𝑚 cmap 6535 ℂcc 7611 < clt 7793 − cmin 7926 ℝ+crp 9434 abscabs 10762 –cn→ccncf 12715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-cncf 12716 |
This theorem is referenced by: cncff 12722 cncfi 12723 rescncf 12726 climcncf 12729 cncfco 12736 cnlimci 12800 |
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