![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnptop1 | 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 |
---|---|
cnptop1 | ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid 2798 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1, 2 | iscnp2 21844 | . . 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 501 | . 2 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ ∪ 𝐽)) |
5 | 4 | simp1d 1139 | 1 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 ∪ cuni 4800 “ cima 5522 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Topctop 21498 CnP ccnp 21830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-top 21499 df-topon 21516 df-cnp 21833 |
This theorem is referenced by: cnpco 21872 cncnp2 21886 cnpresti 21893 cnprest2 21895 lmcnp 21909 |
Copyright terms: Public domain | W3C validator |