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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 20854 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ Top) |
3 | cnvimass 5404 | . . 3 ⊢ (◡𝐹 “ 𝑆) ⊆ dom 𝐹 | |
4 | eqid 2610 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
5 | cncls2i.1 | . . . . . 6 ⊢ 𝑌 = ∪ 𝐾 | |
6 | 4, 5 | cnf 20860 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌) |
7 | fdm 5964 | . . . . 5 ⊢ (𝐹:∪ 𝐽⟶𝑌 → dom 𝐹 = ∪ 𝐽) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → dom 𝐹 = ∪ 𝐽) |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → dom 𝐹 = ∪ 𝐽) |
10 | 3, 9 | syl5sseq 3616 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ 𝑆) ⊆ ∪ 𝐽) |
11 | cntop2 20855 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
12 | 5 | ntropn 20663 | . . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ∈ 𝐾) |
13 | 11, 12 | sylan 487 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ∈ 𝐾) |
14 | cnima 20879 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ ((int‘𝐾)‘𝑆) ∈ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ∈ 𝐽) | |
15 | 13, 14 | syldan 486 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ∈ 𝐽) |
16 | 5 | ntrss2 20671 | . . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ⊆ 𝑆) |
17 | 11, 16 | sylan 487 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ⊆ 𝑆) |
18 | imass2 5420 | . . 3 ⊢ (((int‘𝐾)‘𝑆) ⊆ 𝑆 → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ (◡𝐹 “ 𝑆)) | |
19 | 17, 18 | syl 17 | . 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ (◡𝐹 “ 𝑆)) |
20 | 4 | ssntr 20672 | . 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))) |
21 | 2, 10, 15, 19, 20 | syl22anc 1319 | 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 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 ◡ccnv 5037 dom cdm 5038 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Topctop 20517 intcnt 20631 Cn ccn 20838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-top 20521 df-topon 20523 df-ntr 20634 df-cn 20841 |
This theorem is referenced by: cnntr 20889 hmeontr 21382 |
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