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Mirrors > Home > MPE Home > Th. List > hmeocls | Structured version Visualization version GIF version |
Description: Homeomorphisms preserve closures. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
hmeoopn.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
hmeocls | ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocnvcn 22366 | . . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) | |
2 | hmeoopn.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | cncls2i 21875 | . . . 4 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(◡◡𝐹 “ 𝐴)) ⊆ (◡◡𝐹 “ ((cls‘𝐽)‘𝐴))) |
4 | 1, 3 | sylan 583 | . . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(◡◡𝐹 “ 𝐴)) ⊆ (◡◡𝐹 “ ((cls‘𝐽)‘𝐴))) |
5 | imacnvcnv 6030 | . . . 4 ⊢ (◡◡𝐹 “ 𝐴) = (𝐹 “ 𝐴) | |
6 | 5 | fveq2i 6648 | . . 3 ⊢ ((cls‘𝐾)‘(◡◡𝐹 “ 𝐴)) = ((cls‘𝐾)‘(𝐹 “ 𝐴)) |
7 | imacnvcnv 6030 | . . 3 ⊢ (◡◡𝐹 “ ((cls‘𝐽)‘𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴)) | |
8 | 4, 6, 7 | 3sstr3g 3959 | . 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(𝐹 “ 𝐴)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝐴))) |
9 | hmeocn 22365 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
10 | 2 | cnclsi 21877 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝐴))) |
11 | 9, 10 | sylan 583 | . 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝐴))) |
12 | 8, 11 | eqssd 3932 | 1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∪ cuni 4800 ◡ccnv 5518 “ cima 5522 ‘cfv 6324 (class class class)co 7135 clsccl 21623 Cn ccn 21829 Homeochmeo 22358 |
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-rep 5154 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-reu 3113 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-int 4839 df-iun 4883 df-iin 4884 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-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-top 21499 df-topon 21516 df-cld 21624 df-cls 21626 df-cn 21832 df-hmeo 22360 |
This theorem is referenced by: reghmph 22398 nrmhmph 22399 snclseqg 22721 |
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