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Mirrors > Home > MPE Home > Th. List > cnptop2 | Structured version Visualization version GIF version |
Description: Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnptop2 | ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2725 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | iscnp2 23192 | . . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ ∪ 𝐽) ∧ (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))))) |
4 | 3 | simplbi 496 | . 2 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ ∪ 𝐽)) |
5 | 4 | simp2d 1140 | 1 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 ⊆ wss 3944 ∪ cuni 4909 “ cima 5681 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 Topctop 22844 CnP ccnp 23178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-map 8847 df-top 22845 df-topon 22862 df-cnp 23181 |
This theorem is referenced by: cnpco 23220 cncnp2 23234 cnpresti 23241 cnprest 23242 lmcnp 23257 cnpflfi 23952 flfcnp 23957 flfcnp2 23960 |
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