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Mirrors > Home > MPE Home > Th. List > cnntri | Structured version Visualization version GIF version |
Description: Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
cncls2i.1 | ⊢ 𝑌 = ∪ 𝐾 |
Ref | Expression |
---|---|
cnntri | ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntop1 21842 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
2 | 1 | adantr 483 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ Top) |
3 | cnvimass 5943 | . . 3 ⊢ (◡𝐹 “ 𝑆) ⊆ dom 𝐹 | |
4 | eqid 2821 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
5 | cncls2i.1 | . . . . . 6 ⊢ 𝑌 = ∪ 𝐾 | |
6 | 4, 5 | cnf 21848 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌) |
7 | 6 | fdmd 6517 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → dom 𝐹 = ∪ 𝐽) |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → dom 𝐹 = ∪ 𝐽) |
9 | 3, 8 | sseqtrid 4018 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ 𝑆) ⊆ ∪ 𝐽) |
10 | cntop2 21843 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
11 | 5 | ntropn 21651 | . . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ∈ 𝐾) |
12 | 10, 11 | sylan 582 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ∈ 𝐾) |
13 | cnima 21867 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ((int‘𝐾)‘𝑆) ∈ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ∈ 𝐽) | |
14 | 12, 13 | syldan 593 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ∈ 𝐽) |
15 | 5 | ntrss2 21659 | . . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ⊆ 𝑆) |
16 | 10, 15 | sylan 582 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ⊆ 𝑆) |
17 | imass2 5959 | . . 3 ⊢ (((int‘𝐾)‘𝑆) ⊆ 𝑆 → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ (◡𝐹 “ 𝑆)) | |
18 | 16, 17 | syl 17 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ (◡𝐹 “ 𝑆)) |
19 | 4 | ssntr 21660 | . 2 ⊢ (((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑆) ⊆ ∪ 𝐽) ∧ ((◡𝐹 “ ((int‘𝐾)‘𝑆)) ∈ 𝐽 ∧ (◡𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ (◡𝐹 “ 𝑆))) → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑆))) |
20 | 2, 9, 14, 18, 19 | syl22anc 836 | 1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ∪ cuni 4831 ◡ccnv 5548 dom cdm 5549 “ cima 5552 ‘cfv 6349 (class class class)co 7150 Topctop 21495 intcnt 21619 Cn ccn 21826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-top 21496 df-topon 21513 df-ntr 21622 df-cn 21829 |
This theorem is referenced by: cnntr 21877 hmeontr 22371 |
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