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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 14441 | . . 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 6087 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ∈ 𝒫 ℂ) |
3 | 2 | elpwid 3600 | 1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2159 ∀wral 2467 ∃wrex 2468 {crab 2471 ⊆ wss 3143 𝒫 cpw 3589 class class class wbr 4017 ‘cfv 5230 (class class class)co 5890 ↑𝑚 cmap 6665 ℂcc 7826 < clt 8009 − cmin 8145 ℝ+crp 9670 abscabs 11023 –cn→ccncf 14440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ral 2472 df-rex 2473 df-v 2753 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-br 4018 df-opab 4079 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-iota 5192 df-fun 5232 df-fv 5238 df-ov 5893 df-oprab 5894 df-mpo 5895 df-cncf 14441 |
This theorem is referenced by: cncff 14447 cncfi 14448 rescncf 14451 climcncf 14454 cncfco 14461 cnlimci 14525 |
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